CS70: Jean Walrand: Lecture 22. How to model uncertainty? CS70: - - PowerPoint PPT Presentation

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CS70: Jean Walrand: Lecture 22. How to model uncertainty? CS70: - - PowerPoint PPT Presentation

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CS70: Jean Walrand: Lecture 22. How to model uncertainty? CS70: Jean Walrand: Lecture 22. How to model uncertainty? 1. Why Probability? 2. Applications. 3. Sample Spaces. 4. Examples. Why Probability? Why Probability? Two aspects of

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CS70: Jean Walrand: Lecture 22.

How to model uncertainty?

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CS70: Jean Walrand: Lecture 22.

How to model uncertainty?

  • 1. Why Probability?
  • 2. Applications.
  • 3. Sample Spaces.
  • 4. Examples.
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SLIDE 3

Why Probability?

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SLIDE 4

Why Probability?

◮ Two aspects of probability:

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Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability

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Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

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SLIDE 7

Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

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SLIDE 8

Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather

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SLIDE 9

Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather ◮ Stock market

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SLIDE 10

Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather ◮ Stock market ◮ Noise and corruption of signals your cell phone receives

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SLIDE 11

Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather ◮ Stock market ◮ Noise and corruption of signals your cell phone receives ◮ Diseases, etc.

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Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather ◮ Stock market ◮ Noise and corruption of signals your cell phone receives ◮ Diseases, etc.

◮ Examples of man-made randomness:

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Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather ◮ Stock market ◮ Noise and corruption of signals your cell phone receives ◮ Diseases, etc.

◮ Examples of man-made randomness:

◮ Play rock-paper-scissors,

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Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather ◮ Stock market ◮ Noise and corruption of signals your cell phone receives ◮ Diseases, etc.

◮ Examples of man-made randomness:

◮ Play rock-paper-scissors, or tennis, etc.

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Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather ◮ Stock market ◮ Noise and corruption of signals your cell phone receives ◮ Diseases, etc.

◮ Examples of man-made randomness:

◮ Play rock-paper-scissors, or tennis, etc. ◮ WiFi algorithm: randomized

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Why Probability?

◮ Two aspects of probability:

◮ Model key feature of life: Unpredictability ◮ Man-made randomness: Randomized algorithms

◮ Examples of unpredictability:

◮ Weather ◮ Stock market ◮ Noise and corruption of signals your cell phone receives ◮ Diseases, etc.

◮ Examples of man-made randomness:

◮ Play rock-paper-scissors, or tennis, etc. ◮ WiFi algorithm: randomized ◮ Randomized Quicksort

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Key insight

◮ Unpredictable does not mean “nothing is known”

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist]

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

◮ Buy stocks

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

◮ Buy stocks ◮ Detect signals (transmitted bits, speech, images, radar,

diseases, etc.)

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

◮ Buy stocks ◮ Detect signals (transmitted bits, speech, images, radar,

diseases, etc.)

◮ Control systems (Internet, airplane, robots, self-driving

cars, etc.)

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

◮ Buy stocks ◮ Detect signals (transmitted bits, speech, images, radar,

diseases, etc.)

◮ Control systems (Internet, airplane, robots, self-driving

cars, etc.)

◮ How to best use ‘artificial’ uncertainty?

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

◮ Buy stocks ◮ Detect signals (transmitted bits, speech, images, radar,

diseases, etc.)

◮ Control systems (Internet, airplane, robots, self-driving

cars, etc.)

◮ How to best use ‘artificial’ uncertainty?

◮ Play games of chance

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

◮ Buy stocks ◮ Detect signals (transmitted bits, speech, images, radar,

diseases, etc.)

◮ Control systems (Internet, airplane, robots, self-driving

cars, etc.)

◮ How to best use ‘artificial’ uncertainty?

◮ Play games of chance ◮ Design randomized algorithms.

◮ Probability

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

◮ Buy stocks ◮ Detect signals (transmitted bits, speech, images, radar,

diseases, etc.)

◮ Control systems (Internet, airplane, robots, self-driving

cars, etc.)

◮ How to best use ‘artificial’ uncertainty?

◮ Play games of chance ◮ Design randomized algorithms.

◮ Probability

◮ Models knowledge about uncertainty

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Key insight

◮ Unpredictable does not mean “nothing is known”

◮ Coin flip: 50% chance of ‘tails’ [subjectivist] ◮ Many coin flips: About half yield ‘tails’ [frequentist]

◮ How to best make decisions under uncertainty?

◮ Buy stocks ◮ Detect signals (transmitted bits, speech, images, radar,

diseases, etc.)

◮ Control systems (Internet, airplane, robots, self-driving

cars, etc.)

◮ How to best use ‘artificial’ uncertainty?

◮ Play games of chance ◮ Design randomized algorithms.

◮ Probability

◮ Models knowledge about uncertainty ◮ Discovers best way to use that knowledge in making

decisions

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Applications

Some applications

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Applications

Some applications (they are everywhere ...):

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Applications

Some applications (they are everywhere ...):

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Applications

Some applications (they are everywhere ...):

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Applications

Some applications (they are everywhere ...):

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SLIDE 36

Applications

Some applications (they are everywhere ...):

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Probability.

◮ The probability of getting a straight is around 1 in 250.

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Probability.

◮ The probability of getting a straight is around 1 in 250.

Straight: Consecutive cards, suit doesn’t matter, e.g.,

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Probability.

◮ The probability of getting a straight is around 1 in 250.

Straight: Consecutive cards, suit doesn’t matter, e.g.,

◮ The probability of rolling snake eyes is 1/36.

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Probability.

◮ The probability of getting a straight is around 1 in 250.

Straight: Consecutive cards, suit doesn’t matter, e.g.,

◮ The probability of rolling snake eyes is 1/36. How many

snake eyes? one pip and one pip.

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Probability.

◮ The probability of getting a straight is around 1 in 250.

Straight: Consecutive cards, suit doesn’t matter, e.g.,

◮ The probability of rolling snake eyes is 1/36. How many

snake eyes? one pip and one pip. 1∗1 = 1.

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Probability.

◮ The probability of getting a straight is around 1 in 250.

Straight: Consecutive cards, suit doesn’t matter, e.g.,

◮ The probability of rolling snake eyes is 1/36. How many

snake eyes? one pip and one pip. 1∗1 = 1.

◮ The probability that a poll of a 1000 people will report at

least 50% support for a candidate with 60% support is 80%.

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Terminology

◮ “Heads or Tails” in coin tossing (or coin flipping).

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Terminology

◮ “Heads or Tails” in coin tossing (or coin flipping).

◮ One side of a coin is called the “heads” side and the other

the “tails” side.

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Terminology

◮ “Heads or Tails” in coin tossing (or coin flipping).

◮ One side of a coin is called the “heads” side and the other

the “tails” side.

◮ Thus, one says that one gets one heads or one tails in a

coin flip,

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Terminology

◮ “Heads or Tails” in coin tossing (or coin flipping).

◮ One side of a coin is called the “heads” side and the other

the “tails” side.

◮ Thus, one says that one gets one heads or one tails in a

coin flip, or that the coin flip is heads or tails.

◮ Rolling one die or two dice.

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Terminology

◮ “Heads or Tails” in coin tossing (or coin flipping).

◮ One side of a coin is called the “heads” side and the other

the “tails” side.

◮ Thus, one says that one gets one heads or one tails in a

coin flip, or that the coin flip is heads or tails.

◮ Rolling one die or two dice.

◮ Singular is die and plural is dice.

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Probability

◮ If you flip a fair coin and get 50 heads, you will get heads

the next time with probability ...

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Probability

◮ If you flip a fair coin and get 50 heads, you will get heads

the next time with probability ...1/2.

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Probability

◮ If you flip a fair coin and get 50 heads, you will get heads

the next time with probability ...1/2.

◮ The probability that the next person through the door is

younger than 21 is 80%.

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Probability

◮ If you flip a fair coin and get 50 heads, you will get heads

the next time with probability ...1/2.

◮ The probability that the next person through the door is

younger than 21 is 80%.

◮ Amanda Knox is innocent with 70% probability.

They are statements about a probability space.

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Probability

◮ If you flip a fair coin and get 50 heads, you will get heads

the next time with probability ...1/2.

◮ The probability that the next person through the door is

younger than 21 is 80%.

◮ Amanda Knox is innocent with 70% probability.

They are statements about a probability space. (Except perhaps the last one or two.)

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Probability

◮ If you flip a fair coin and get 50 heads, you will get heads

the next time with probability ...1/2.

◮ The probability that the next person through the door is

younger than 21 is 80%.

◮ Amanda Knox is innocent with 70% probability.

They are statements about a probability space. (Except perhaps the last one or two.) Random experiment constructed by us, or the world.

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Probability Space.

  • 1. A “random experiment”:
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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin;

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins;

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.
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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T};

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T}; (b) Ω = {HH,HT,TH,TT};

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T}; (b) Ω = {HH,HT,TH,TT}; |Ω| =

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T}; (b) Ω = {HH,HT,TH,TT}; |Ω| = 4;

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T}; (b) Ω = {HH,HT,TH,TT}; |Ω| = 4; (c) Ω = { A♠ A♦ A♣ A♥ K♠, A♠ A♦ A♣ A♥ Q♠,...} |Ω| =

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T}; (b) Ω = {HH,HT,TH,TT}; |Ω| = 4; (c) Ω = { A♠ A♦ A♣ A♥ K♠, A♠ A♦ A♣ A♥ Q♠,...} |Ω| = 52

5

  • .
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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T}; (b) Ω = {HH,HT,TH,TT}; |Ω| = 4; (c) Ω = { A♠ A♦ A♣ A♥ K♠, A♠ A♦ A♣ A♥ Q♠,...} |Ω| = 52

5

  • .
  • 3. Assign a probability to each outcome: Pr : Ω → [0,1].

(a) Pr[H] = p,Pr[T] = 1−p for some p ∈ [0,1]

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T}; (b) Ω = {HH,HT,TH,TT}; |Ω| = 4; (c) Ω = { A♠ A♦ A♣ A♥ K♠, A♠ A♦ A♣ A♥ Q♠,...} |Ω| = 52

5

  • .
  • 3. Assign a probability to each outcome: Pr : Ω → [0,1].

(a) Pr[H] = p,Pr[T] = 1−p for some p ∈ [0,1] (b) Pr[HH] = Pr[HT] = Pr[TH] = Pr[TT] = 1

4

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Probability Space.

  • 1. A “random experiment”:

(a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.

  • 2. A set of possible outcomes: Ω.

(a) Ω = {H,T}; (b) Ω = {HH,HT,TH,TT}; |Ω| = 4; (c) Ω = { A♠ A♦ A♣ A♥ K♠, A♠ A♦ A♣ A♥ Q♠,...} |Ω| = 52

5

  • .
  • 3. Assign a probability to each outcome: Pr : Ω → [0,1].

(a) Pr[H] = p,Pr[T] = 1−p for some p ∈ [0,1] (b) Pr[HH] = Pr[HT] = Pr[TH] = Pr[TT] = 1

4

(c) Pr[ A♠ A♦ A♣ A♥ K♠ ] = ··· = 1/ 52

5

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Probability Space: formalism.

Ω is the sample space.

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Probability Space: formalism.

Ω is the sample space. ω ∈ Ω is a sample point.

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Probability Space: formalism.

Ω is the sample space. ω ∈ Ω is a sample point. (Also called an outcome.)

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Probability Space: formalism.

Ω is the sample space. ω ∈ Ω is a sample point. (Also called an outcome.) Sample point ω has a probability Pr[ω] where

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Probability Space: formalism.

Ω is the sample space. ω ∈ Ω is a sample point. (Also called an outcome.) Sample point ω has a probability Pr[ω] where

◮ 0 ≤ Pr[ω] ≤ 1;

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Probability Space: formalism.

Ω is the sample space. ω ∈ Ω is a sample point. (Also called an outcome.) Sample point ω has a probability Pr[ω] where

◮ 0 ≤ Pr[ω] ≤ 1; ◮ ∑ω∈Ω Pr[ω] = 1.

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Probability Space: formalism.

Ω is the sample space. ω ∈ Ω is a sample point. (Also called an outcome.) Sample point ω has a probability Pr[ω] where

◮ 0 ≤ Pr[ω] ≤ 1; ◮ ∑ω∈Ω Pr[ω] = 1.

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Probability Space: Formalism.

In a uniform probability space each outcome ω is equally probable: Pr[ω] =

1 |Ω| for all ω ∈ Ω.

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SLIDE 76

Probability Space: Formalism.

In a uniform probability space each outcome ω is equally probable: Pr[ω] =

1 |Ω| for all ω ∈ Ω.

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SLIDE 77

Probability Space: Formalism.

In a uniform probability space each outcome ω is equally probable: Pr[ω] =

1 |Ω| for all ω ∈ Ω.

Examples:

◮ Flipping two fair coins, dealing a poker hand are uniform

probability spaces.

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SLIDE 78

Probability Space: Formalism.

In a uniform probability space each outcome ω is equally probable: Pr[ω] =

1 |Ω| for all ω ∈ Ω.

Examples:

◮ Flipping two fair coins, dealing a poker hand are uniform

probability spaces.

◮ Flipping a biased coin is not a uniform probability space.

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Probability Space: Formalism

Simplest physical model of a uniform probability space:

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Probability Space: Formalism

Simplest physical model of a uniform probability space:

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Probability Space: Formalism

Simplest physical model of a uniform probability space: A bag of identical balls, except for their color (or a label).

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Probability Space: Formalism

Simplest physical model of a uniform probability space: A bag of identical balls, except for their color (or a label). If the bag is well shaken, every ball is equally likely to be picked.

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SLIDE 83

Probability Space: Formalism

Simplest physical model of a non-uniform probability space:

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Probability Space: Formalism

Simplest physical model of a non-uniform probability space:

p3 Fraction p1

  • f circumference

p2

pω ω

1 2 3

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SLIDE 85

Probability Space: Formalism

Simplest physical model of a non-uniform probability space:

p3 Fraction p1

  • f circumference

p2

pω ω

1 2 3

The roulette wheel stops in sector ω with probability pω.

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SLIDE 86

Probability Space: Formalism

Simplest physical model of a non-uniform probability space:

p3 Fraction p1

  • f circumference

p2

pω ω

1 2 3

The roulette wheel stops in sector ω with probability pω. (Imagine a perfectly constructed wheel, that you spin hard enough, with low friction... .)

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SLIDE 87

Probability of exactly one heads in two coin flips?

Idea: Sum the probabilities of the outcomes with one heads.

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SLIDE 88

Probability of exactly one heads in two coin flips?

Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition!

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SLIDE 89

Probability of exactly one heads in two coin flips?

Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition! An event, E, is a subset of outcomes.

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SLIDE 90

Probability of exactly one heads in two coin flips?

Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition! An event, E, is a subset of outcomes. Pr[E] = ∑ω∈E Pr[ω].

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SLIDE 91

Probability of exactly one heads in two coin flips?

Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition! An event, E, is a subset of outcomes. Pr[E] = ∑ω∈E Pr[ω].

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SLIDE 92

Probability of exactly one heads in two coin flips?

Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition! An event, E, is a subset of outcomes. Pr[E] = ∑ω∈E Pr[ω].

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Event

p3 p2

pω ω

1 2 3 p1

E

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SLIDE 94

Event

p3 p2

pω ω

1 2 3 p1

E

Event E = ‘wheel stops in sector 1 or 2 or 3’

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SLIDE 95

Event

p3 p2

pω ω

1 2 3 p1

E

Event E = ‘wheel stops in sector 1 or 2 or 3’ = {1,2,3}.

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SLIDE 96

Event

p3 p2

pω ω

1 2 3 p1

E

Event E = ‘wheel stops in sector 1 or 2 or 3’ = {1,2,3}. Pr[E] = p1 +p2 +p3

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SLIDE 97

Probability of exactly one heads in two coin flips?

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SLIDE 98

Probability of exactly one heads in two coin flips?

Sample Space, Ω = {HH,HT,TH,TT}.

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SLIDE 99

Probability of exactly one heads in two coin flips?

Sample Space, Ω = {HH,HT,TH,TT}. Uniform probability space: Pr[HH] = Pr[HT] = Pr[TH] = Pr[TT] = 1

4.

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SLIDE 100

Probability of exactly one heads in two coin flips?

Sample Space, Ω = {HH,HT,TH,TT}. Uniform probability space: Pr[HH] = Pr[HT] = Pr[TH] = Pr[TT] = 1

4.

Event, E, “exactly one heads”: {TH,HT}.

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SLIDE 101

Probability of exactly one heads in two coin flips?

Sample Space, Ω = {HH,HT,TH,TT}. Uniform probability space: Pr[HH] = Pr[HT] = Pr[TH] = Pr[TT] = 1

4.

Event, E, “exactly one heads”: {TH,HT}. Pr[E] = ∑

ω∈E

Pr[ω]

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SLIDE 102

Probability of exactly one heads in two coin flips?

Sample Space, Ω = {HH,HT,TH,TT}. Uniform probability space: Pr[HH] = Pr[HT] = Pr[TH] = Pr[TT] = 1

4.

Event, E, “exactly one heads”: {TH,HT}. Pr[E] = ∑

ω∈E

Pr[ω] = |E| |Ω|

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SLIDE 103

Probability of exactly one heads in two coin flips?

Sample Space, Ω = {HH,HT,TH,TT}. Uniform probability space: Pr[HH] = Pr[HT] = Pr[TH] = Pr[TT] = 1

4.

Event, E, “exactly one heads”: {TH,HT}. Pr[E] = ∑

ω∈E

Pr[ω] = |E| |Ω| = 2 4

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SLIDE 104

Probability of exactly one heads in two coin flips?

Sample Space, Ω = {HH,HT,TH,TT}. Uniform probability space: Pr[HH] = Pr[HT] = Pr[TH] = Pr[TT] = 1

4.

Event, E, “exactly one heads”: {TH,HT}. Pr[E] = ∑

ω∈E

Pr[ω] = |E| |Ω| = 2 4 = 1 2.

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SLIDE 105

Probability of a straight?

Recall: Straight := Consecutive cards, suit does not matter. Outcomes: Ω = “poker hands”.

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SLIDE 106

Probability of a straight?

Recall: Straight := Consecutive cards, suit does not matter. Outcomes: Ω = “poker hands”. Uniform probability space Pr[ω] = 1 |Ω| = 1 52

5

.

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SLIDE 107

Probability of a straight?

Recall: Straight := Consecutive cards, suit does not matter. Outcomes: Ω = “poker hands”. Uniform probability space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a straight }.

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SLIDE 108

Probability of a straight?

Recall: Straight := Consecutive cards, suit does not matter. Outcomes: Ω = “poker hands”. Uniform probability space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a straight }.

∑

ω∈E

Pr[ω] = |E| |Ω|.

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SLIDE 109

Construct straight:

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SLIDE 110

Construct straight:

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SLIDE 111

Construct straight: First choose the smallest value of the cards: {A,...,10} :

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SLIDE 112

Construct straight: First choose the smallest value of the cards: {A,...,10} : 10 ways

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SLIDE 113

Construct straight: First choose the smallest value of the cards: {A,...,10} : 10 ways and then five choices of suit:

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SLIDE 114

Construct straight: First choose the smallest value of the cards: {A,...,10} : 10 ways and then five choices of suit: 5 choices, 4 ways for each

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SLIDE 115

Construct straight: First choose the smallest value of the cards: {A,...,10} : 10 ways and then five choices of suit: 5 choices, 4 ways for each |E| = 10×4×4×4×4×4

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SLIDE 116

Construct straight: First choose the smallest value of the cards: {A,...,10} : 10 ways and then five choices of suit: 5 choices, 4 ways for each |E| = 10×4×4×4×4×4 = 10×(45)

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SLIDE 117

Construct straight: First choose the smallest value of the cards: {A,...,10} : 10 ways and then five choices of suit: 5 choices, 4 ways for each |E| = 10×4×4×4×4×4 = 10×(45) Pr[E] = ∑

ω∈E

Pr[ω]

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SLIDE 118

Construct straight: First choose the smallest value of the cards: {A,...,10} : 10 ways and then five choices of suit: 5 choices, 4 ways for each |E| = 10×4×4×4×4×4 = 10×(45) Pr[E] = ∑

ω∈E

Pr[ω] = 10∗45 52

5

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SLIDE 119

Calculation. Pr[E] = 10∗45 52

5

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SLIDE 120

Calculation. Pr[E] = 10∗45 52

5

  • irb(main):004:0*> 52*51*50*49*48/((5*4*3*2)*10*4**5)
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SLIDE 121

Calculation. Pr[E] = 10∗45 52

5

  • irb(main):004:0*> 52*51*50*49*48/((5*4*3*2)*10*4**5)

=> 253

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SLIDE 122

Calculation. Pr[E] = 10∗45 52

5

  • irb(main):004:0*> 52*51*50*49*48/((5*4*3*2)*10*4**5)

=> 253

Thus, Pr[straight] ≈ 1 253.

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SLIDE 123

Is a flush more likely than a straight?

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SLIDE 124

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit.

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SLIDE 125

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”.

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SLIDE 126

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

.

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SLIDE 127

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

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SLIDE 128

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|.

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SLIDE 129

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|?

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SLIDE 130

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush:

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SLIDE 131

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush: First choose the suit –

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SLIDE 132

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush: First choose the suit – 4 ways

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SLIDE 133

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush: First choose the suit – 4 ways and then choose five cards from 13:

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SLIDE 134

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush: First choose the suit – 4 ways and then choose five cards from 13: 13

5

  • ways
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SLIDE 135

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush: First choose the suit – 4 ways and then choose five cards from 13: 13

5

  • ways

|E| = 4× 13 5

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SLIDE 136

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush: First choose the suit – 4 ways and then choose five cards from 13: 13

5

  • ways

|E| = 4× 13 5

  • Plug in.
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SLIDE 137

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush: First choose the suit – 4 ways and then choose five cards from 13: 13

5

  • ways

|E| = 4× 13 5

  • Plug in.

Pr[ω] = |E| |Ω|

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SLIDE 138

Is a flush more likely than a straight?

A flush is a hand that contains five cards of the same suit. Outcomes: Ω = “poker hands”. Uniform probablity space Pr[ω] = 1 |Ω| = 1 52

5

. Event E = { a flush }.

∑

ω∈E

Pr[ω] = |E| |Ω|. |E|? Construct flush: First choose the suit – 4 ways and then choose five cards from 13: 13

5

  • ways

|E| = 4× 13 5

  • Plug in.

Pr[ω] = |E| |Ω| = 4 13

5

  • 52

5

.

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SLIDE 139

Calculation.

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SLIDE 140

Calculation. Pr[E] = 4 13

5

  • 52

5

  • irb(main):001:0> 52*51*50*49*48/(4*13*12*11*10*9)
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SLIDE 141

Calculation. Pr[E] = 4 13

5

  • 52

5

  • irb(main):001:0> 52*51*50*49*48/(4*13*12*11*10*9)

=> 504

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SLIDE 142

Calculation. Pr[E] = 4 13

5

  • 52

5

  • irb(main):001:0> 52*51*50*49*48/(4*13*12*11*10*9)

=> 504

Hence, Pr[flush] ≈ 1 504.

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SLIDE 143

Calculation. Pr[E] = 4 13

5

  • 52

5

  • irb(main):001:0> 52*51*50*49*48/(4*13*12*11*10*9)

=> 504

Hence, Pr[flush] ≈ 1 504. Thus, a straight is about twice as likely as a flush. (1/253 vs. 1/504.)

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SLIDE 144

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses

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SLIDE 145

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2

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SLIDE 146

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

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SLIDE 147

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

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SLIDE 148

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or

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SLIDE 149

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

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SLIDE 150

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer:

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SLIDE 151

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|.

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SLIDE 152

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|. ◮ What is more likely?

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SLIDE 153

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|. ◮ What is more likely?

(E1) Twenty Hs out of twenty, or

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SLIDE 154

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|. ◮ What is more likely?

(E1) Twenty Hs out of twenty, or (E2) Ten Hs out of twenty?

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SLIDE 155

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|. ◮ What is more likely?

(E1) Twenty Hs out of twenty, or (E2) Ten Hs out of twenty?

Answer: Ten Hs out of twenty. Why?

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SLIDE 156

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|. ◮ What is more likely?

(E1) Twenty Hs out of twenty, or (E2) Ten Hs out of twenty?

Answer: Ten Hs out of twenty. Why? There are many sequences of 20 tosses with ten Hs;

  • nly one with twenty Hs. ⇒ Pr[E1] =

1 |Ω| ≪ Pr[E2] = |E2| |Ω| .

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SLIDE 157

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|. ◮ What is more likely?

(E1) Twenty Hs out of twenty, or (E2) Ten Hs out of twenty?

Answer: Ten Hs out of twenty. Why? There are many sequences of 20 tosses with ten Hs;

  • nly one with twenty Hs. ⇒ Pr[E1] =

1 |Ω| ≪ Pr[E2] = |E2| |Ω| .

|E2| =

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SLIDE 158

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|. ◮ What is more likely?

(E1) Twenty Hs out of twenty, or (E2) Ten Hs out of twenty?

Answer: Ten Hs out of twenty. Why? There are many sequences of 20 tosses with ten Hs;

  • nly one with twenty Hs. ⇒ Pr[E1] =

1 |Ω| ≪ Pr[E2] = |E2| |Ω| .

|E2| = 20 10

  • =
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SLIDE 159

20 coin tosses

Sample space: Ω = set of 20 fair coin tosses = {H,T}20. |Ω| = 2×2×···×2 = 220.

◮ What is more likely?

◮ ω1 := HHHHHHHHHHHHHHHHHHHH, or ◮ ω2 := HHTHTTHHTTHTHHTTHTHT?

Answer: Both are equally likely: Pr[ω1] = Pr[ω2] =

1 |Ω|. ◮ What is more likely?

(E1) Twenty Hs out of twenty, or (E2) Ten Hs out of twenty?

Answer: Ten Hs out of twenty. Why? There are many sequences of 20 tosses with ten Hs;

  • nly one with twenty Hs. ⇒ Pr[E1] =

1 |Ω| ≪ Pr[E2] = |E2| |Ω| .

|E2| = 20 10

  • = 184,756.
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SLIDE 160

Summary

How to model uncertainty?

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SLIDE 161

Summary

How to model uncertainty? Key ideas:

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SLIDE 162

Summary

How to model uncertainty? Key ideas:

◮ Random experiment

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SLIDE 163

Summary

How to model uncertainty? Key ideas:

◮ Random experiment ◮ Probability space

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SLIDE 164

Summary

How to model uncertainty? Key ideas:

◮ Random experiment ◮ Probability space

◮ Sample space Ω

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SLIDE 165

Summary

How to model uncertainty? Key ideas:

◮ Random experiment ◮ Probability space

◮ Sample space Ω ◮ Probability: Pr(ω)

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SLIDE 166

Summary

How to model uncertainty? Key ideas:

◮ Random experiment ◮ Probability space

◮ Sample space Ω ◮ Probability: Pr(ω) ◮ Event: E ⊆ Ω;Pr[E].