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Roulette The game of roulette serves as a nice introduction to some - - PowerPoint PPT Presentation

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Roulette The game of roulette serves as a nice introduction to some of the key ideas relating to probability. Alan H. SteinUniversity of Connecticut Roulette The game of roulette serves as a nice introduction to some of the key ideas relating to

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SLIDE 1

Roulette

The game of roulette serves as a nice introduction to some of the key ideas relating to probability.

Alan H. SteinUniversity of Connecticut

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SLIDE 2

Roulette

The game of roulette serves as a nice introduction to some of the key ideas relating to probability. We will look at some of the possible outcomes when a roulette wheel is spun and relate those

  • utcomes to probability in general.

Alan H. SteinUniversity of Connecticut

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SLIDE 3

Roulette

The game of roulette serves as a nice introduction to some of the key ideas relating to probability. We will look at some of the possible outcomes when a roulette wheel is spun and relate those

  • utcomes to probability in general.

A roulette wheel contains 38 slots, numbered 0, 00, and 1, 2, 3, . . . 36.

Alan H. SteinUniversity of Connecticut

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SLIDE 4

Roulette

The game of roulette serves as a nice introduction to some of the key ideas relating to probability. We will look at some of the possible outcomes when a roulette wheel is spun and relate those

  • utcomes to probability in general.

A roulette wheel contains 38 slots, numbered 0, 00, and 1, 2, 3, . . . 36. When the wheel is spun, a ball eventually falls into

  • ne of the slots.

Alan H. SteinUniversity of Connecticut

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SLIDE 5

Roulette

The game of roulette serves as a nice introduction to some of the key ideas relating to probability. We will look at some of the possible outcomes when a roulette wheel is spun and relate those

  • utcomes to probability in general.

A roulette wheel contains 38 slots, numbered 0, 00, and 1, 2, 3, . . . 36. When the wheel is spun, a ball eventually falls into

  • ne of the slots. Assuming the wheel is balanced and the slots are

the same size, as is supposed to be the case, there are 38 possible

  • utcomes, each of probability 1

38.

Alan H. SteinUniversity of Connecticut

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SLIDE 6

Roulette

The game of roulette serves as a nice introduction to some of the key ideas relating to probability. We will look at some of the possible outcomes when a roulette wheel is spun and relate those

  • utcomes to probability in general.

A roulette wheel contains 38 slots, numbered 0, 00, and 1, 2, 3, . . . 36. When the wheel is spun, a ball eventually falls into

  • ne of the slots. Assuming the wheel is balanced and the slots are

the same size, as is supposed to be the case, there are 38 possible

  • utcomes, each of probability 1

38. This is an example of an Equiprobable Probability Space.

Alan H. SteinUniversity of Connecticut

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SLIDE 7

Terminology

We consider experiments with a finite set of possible outcomes.

Alan H. SteinUniversity of Connecticut

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Terminology

We consider experiments with a finite set of possible outcomes. We call the set of possible outcomes the sample space

Alan H. SteinUniversity of Connecticut

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SLIDE 9

Terminology

We consider experiments with a finite set of possible outcomes. We call the set of possible outcomes the sample space and the individual outcomes are called sample points.

Alan H. SteinUniversity of Connecticut

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Terminology

We consider experiments with a finite set of possible outcomes. We call the set of possible outcomes the sample space and the individual outcomes are called sample points. A subset of the sample space is referred to as an event. We’ll spend time on events later on.

Alan H. SteinUniversity of Connecticut

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Straight Bets

There is a variety of ways one can bet at roulette. We will see the casino doesn’t care how one bets.

Alan H. SteinUniversity of Connecticut

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Straight Bets

There is a variety of ways one can bet at roulette. We will see the casino doesn’t care how one bets. The simplest bet is a straight bet, where one bets on a specific number and wins, with a payoff at 35-1, if that number comes up.

Alan H. SteinUniversity of Connecticut

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SLIDE 13

Straight Bets

There is a variety of ways one can bet at roulette. We will see the casino doesn’t care how one bets. The simplest bet is a straight bet, where one bets on a specific number and wins, with a payoff at 35-1, if that number comes up. This means that if a player bets $1 and wins, the player will get $36 back, the dollar he bet along with $35 more.

Alan H. SteinUniversity of Connecticut

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Straight Bets

There is a variety of ways one can bet at roulette. We will see the casino doesn’t care how one bets. The simplest bet is a straight bet, where one bets on a specific number and wins, with a payoff at 35-1, if that number comes up. This means that if a player bets $1 and wins, the player will get $36 back, the dollar he bet along with $35 more. In our analyses, let’s assume every bet is for $1.

Alan H. SteinUniversity of Connecticut

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Straight Bets – The Big Picture

Suppose someone repeatedly makes a straight bet for $1. Although the results will vary, on average

Alan H. SteinUniversity of Connecticut

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Straight Bets – The Big Picture

Suppose someone repeatedly makes a straight bet for $1. Although the results will vary, on average the player will win about 1 38 of the time and lose the rest of the time.

Alan H. SteinUniversity of Connecticut

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Straight Bets – The Big Picture

Suppose someone repeatedly makes a straight bet for $1. Although the results will vary, on average the player will win about 1 38 of the time and lose the rest of the time. If he (or she) plays 38 times, he could expect to win about one time, coming out $35 ahead that time,

Alan H. SteinUniversity of Connecticut

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SLIDE 18

Straight Bets – The Big Picture

Suppose someone repeatedly makes a straight bet for $1. Although the results will vary, on average the player will win about 1 38 of the time and lose the rest of the time. If he (or she) plays 38 times, he could expect to win about one time, coming out $35 ahead that time, but losing about 37 times, coming out $1 behind those times,

Alan H. SteinUniversity of Connecticut

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SLIDE 19

Straight Bets – The Big Picture

Suppose someone repeatedly makes a straight bet for $1. Although the results will vary, on average the player will win about 1 38 of the time and lose the rest of the time. If he (or she) plays 38 times, he could expect to win about one time, coming out $35 ahead that time, but losing about 37 times, coming out $1 behind those times, so after playing 38 times he can expect to be about $2 behind.

Alan H. SteinUniversity of Connecticut

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SLIDE 20

Straight Bets – The Big Picture

Suppose someone repeatedly makes a straight bet for $1. Although the results will vary, on average the player will win about 1 38 of the time and lose the rest of the time. If he (or she) plays 38 times, he could expect to win about one time, coming out $35 ahead that time, but losing about 37 times, coming out $1 behind those times, so after playing 38 times he can expect to be about $2 behind. That works out to losing about $ 2 38 = $ 1 19 per bet.

Alan H. SteinUniversity of Connecticut

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Random Variables

Random variables are numerical values associated with outcomes

  • f experiments.

Alan H. SteinUniversity of Connecticut

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Random Variables

Random variables are numerical values associated with outcomes

  • f experiments.

The letter X is the most common symbol used to represent a random variable.

Alan H. SteinUniversity of Connecticut

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SLIDE 23

Random Variables

Random variables are numerical values associated with outcomes

  • f experiments.

The letter X is the most common symbol used to represent a random variable. If we consider a straight bet of $1 to be an experiment and X to be the net winnings for the player,

Alan H. SteinUniversity of Connecticut

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SLIDE 24

Random Variables

Random variables are numerical values associated with outcomes

  • f experiments.

The letter X is the most common symbol used to represent a random variable. If we consider a straight bet of $1 to be an experiment and X to be the net winnings for the player, then X can take on either the value of 35 (if the player wins)

Alan H. SteinUniversity of Connecticut

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SLIDE 25

Random Variables

Random variables are numerical values associated with outcomes

  • f experiments.

The letter X is the most common symbol used to represent a random variable. If we consider a straight bet of $1 to be an experiment and X to be the net winnings for the player, then X can take on either the value of 35 (if the player wins) or −1 (if the player loses).

Alan H. SteinUniversity of Connecticut

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SLIDE 26

Random Variables

Random variables are numerical values associated with outcomes

  • f experiments.

The letter X is the most common symbol used to represent a random variable. If we consider a straight bet of $1 to be an experiment and X to be the net winnings for the player, then X can take on either the value of 35 (if the player wins) or −1 (if the player loses). If we denote the probability X takes on a certain value k by P(X = k),

Alan H. SteinUniversity of Connecticut

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SLIDE 27

Random Variables

Random variables are numerical values associated with outcomes

  • f experiments.

The letter X is the most common symbol used to represent a random variable. If we consider a straight bet of $1 to be an experiment and X to be the net winnings for the player, then X can take on either the value of 35 (if the player wins) or −1 (if the player loses). If we denote the probability X takes on a certain value k by P(X = k), then P(X = 35) = 1 38 and P(X = −1) = 37 38.

Alan H. SteinUniversity of Connecticut

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Mathematical Expectation

Mathematical expectation is essentially what we can expect the average value of a random variable to be close to.

Alan H. SteinUniversity of Connecticut

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Mathematical Expectation

Mathematical expectation is essentially what we can expect the average value of a random variable to be close to. We denote it by E(X).

Alan H. SteinUniversity of Connecticut

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Mathematical Expectation

Mathematical expectation is essentially what we can expect the average value of a random variable to be close to. We denote it by E(X). In this case, we expect E(X) = − 1 19,

Alan H. SteinUniversity of Connecticut

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Mathematical Expectation

Mathematical expectation is essentially what we can expect the average value of a random variable to be close to. We denote it by E(X). In this case, we expect E(X) = − 1 19, since we expect to lose an average of $ 1 19 per bet.

Alan H. SteinUniversity of Connecticut

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Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation.

Alan H. SteinUniversity of Connecticut

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Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took

Alan H. SteinUniversity of Connecticut

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Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2.

Alan H. SteinUniversity of Connecticut

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SLIDE 35

Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2. We then divided by 38 to get −2 38 = − 1 19.

Alan H. SteinUniversity of Connecticut

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SLIDE 36

Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2. We then divided by 38 to get −2 38 = − 1 19. If we unravel the calculation, we can write

Alan H. SteinUniversity of Connecticut

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SLIDE 37

Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2. We then divided by 38 to get −2 38 = − 1 19. If we unravel the calculation, we can write − 1 19 = −2 38 =

Alan H. SteinUniversity of Connecticut

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SLIDE 38

Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2. We then divided by 38 to get −2 38 = − 1 19. If we unravel the calculation, we can write − 1 19 = −2 38 = 35 − 37 · 1 38 =

Alan H. SteinUniversity of Connecticut

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SLIDE 39

Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2. We then divided by 38 to get −2 38 = − 1 19. If we unravel the calculation, we can write − 1 19 = −2 38 = 35 − 37 · 1 38 = 35 · 1 − 1 · 37 38 =

Alan H. SteinUniversity of Connecticut

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SLIDE 40

Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2. We then divided by 38 to get −2 38 = − 1 19. If we unravel the calculation, we can write − 1 19 = −2 38 = 35 − 37 · 1 38 = 35 · 1 − 1 · 37 38 = 35 · 1 + (−1) · 37 38 =

Alan H. SteinUniversity of Connecticut

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SLIDE 41

Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2. We then divided by 38 to get −2 38 = − 1 19. If we unravel the calculation, we can write − 1 19 = −2 38 = 35 − 37 · 1 38 = 35 · 1 − 1 · 37 38 = 35 · 1 + (−1) · 37 38 = 35 · 1 38 + (−1) · 37 38 =

Alan H. SteinUniversity of Connecticut

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SLIDE 42

Calculating Mathematical Expectation

If we look at our earlier calculation, we can infer a reasonable definition of mathematical expectation. Imagining playing 38 times, we took the 1 expected win, with X being 35 that time, the 37 expected losses, with X being −1 those times, and took 35 − 37 · 1 = −2. We then divided by 38 to get −2 38 = − 1 19. If we unravel the calculation, we can write − 1 19 = −2 38 = 35 − 37 · 1 38 = 35 · 1 − 1 · 37 38 = 35 · 1 + (−1) · 37 38 = 35 · 1 38 + (−1) · 37 38 = 35P(X = 35) + (−1)P(X = −1).

Alan H. SteinUniversity of Connecticut

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Abstracting to Get a Formula

The calculation E(X) = 35P(X = 35) + (−1)P(X = −1) suggests we calculate mathematical expectation by taking each possible value of the random variable, multiply it by the probability the random variable takes on that value, and add the products together.

Alan H. SteinUniversity of Connecticut

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Abstracting to Get a Formula

The calculation E(X) = 35P(X = 35) + (−1)P(X = −1) suggests we calculate mathematical expectation by taking each possible value of the random variable, multiply it by the probability the random variable takes on that value, and add the products together. Symbolically, we may write this as

Alan H. SteinUniversity of Connecticut

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Abstracting to Get a Formula

The calculation E(X) = 35P(X = 35) + (−1)P(X = −1) suggests we calculate mathematical expectation by taking each possible value of the random variable, multiply it by the probability the random variable takes on that value, and add the products together. Symbolically, we may write this as E(X) = kP(X = k), where the sum is taken over all the possible values for X.

Alan H. SteinUniversity of Connecticut

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SLIDE 46

Probability Spaces

Recall:

Alan H. SteinUniversity of Connecticut

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SLIDE 47

Probability Spaces

Recall: We consider experiments with a finite set of possible outcomes.

Alan H. SteinUniversity of Connecticut

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SLIDE 48

Probability Spaces

Recall: We consider experiments with a finite set of possible outcomes. We call the set of possible outcomes the sample space

Alan H. SteinUniversity of Connecticut

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SLIDE 49

Probability Spaces

Recall: We consider experiments with a finite set of possible outcomes. We call the set of possible outcomes the sample space and the individual outcomes are called sample points.

Alan H. SteinUniversity of Connecticut

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SLIDE 50

Probability Spaces

Recall: We consider experiments with a finite set of possible outcomes. We call the set of possible outcomes the sample space and the individual outcomes are called sample points. We’re interested in the likelihood of each possibility and assign, hopefully in a meaningful way, a probability to each.

Alan H. SteinUniversity of Connecticut

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SLIDE 51

Probability Spaces

Recall: We consider experiments with a finite set of possible outcomes. We call the set of possible outcomes the sample space and the individual outcomes are called sample points. We’re interested in the likelihood of each possibility and assign, hopefully in a meaningful way, a probability to each.

Definition (Probability)

A probability is a number between 0 and 1.

Alan H. SteinUniversity of Connecticut

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SLIDE 52

Probability Spaces

Recall: We consider experiments with a finite set of possible outcomes. We call the set of possible outcomes the sample space and the individual outcomes are called sample points. We’re interested in the likelihood of each possibility and assign, hopefully in a meaningful way, a probability to each.

Definition (Probability)

A probability is a number between 0 and 1.

Definition (Probability Space)

A probability space is a sample space S for which a probability p(x) has been assigned to each sample point x such that the sum

  • x∈S p(x) of the probabilities is 1.

Alan H. SteinUniversity of Connecticut

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SLIDE 53

Examples

Consider the sample space {H,T}, with probabilities p(H) = p(T) = 1 2.

Alan H. SteinUniversity of Connecticut

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SLIDE 54

Examples

Consider the sample space {H,T}, with probabilities p(H) = p(T) = 1

  • 2. This may be considered a model for flipping a

coin a single time.

Alan H. SteinUniversity of Connecticut

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SLIDE 55

Examples

Consider the sample space {H,T}, with probabilities p(H) = p(T) = 1

  • 2. This may be considered a model for flipping a

coin a single time. Consider the sample space {H0, H1, H2}, with probabilities p(H0) = 1 4, p(H1) = 1 2, p(H2) = 1 4.

Alan H. SteinUniversity of Connecticut

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SLIDE 56

Examples

Consider the sample space {H,T}, with probabilities p(H) = p(T) = 1

  • 2. This may be considered a model for flipping a

coin a single time. Consider the sample space {H0, H1, H2}, with probabilities p(H0) = 1 4, p(H1) = 1 2, p(H2) = 1

  • 4. This may be considered a

model for flipping a coin twice.

Alan H. SteinUniversity of Connecticut

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SLIDE 57

Examples

Consider the sample space {H,T}, with probabilities p(H) = p(T) = 1

  • 2. This may be considered a model for flipping a

coin a single time. Consider the sample space {H0, H1, H2}, with probabilities p(H0) = 1 4, p(H1) = 1 2, p(H2) = 1

  • 4. This may be considered a

model for flipping a coin twice. The first example is known as an Equiprobable Space; the second is not an equiprobable space.

Alan H. SteinUniversity of Connecticut

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SLIDE 58

Equiprobable Spaces

Definition (Equiprobable Space)

A probability space P is called an equiprobable space if each sample point has the same probability.

Alan H. SteinUniversity of Connecticut

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SLIDE 59

Equiprobable Spaces

Definition (Equiprobable Space)

A probability space P is called an equiprobable space if each sample point has the same probability. If an equiprobabe space P has n points,

Alan H. SteinUniversity of Connecticut

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SLIDE 60

Equiprobable Spaces

Definition (Equiprobable Space)

A probability space P is called an equiprobable space if each sample point has the same probability. If an equiprobabe space P has n points, we’ll write |P| = n,

Alan H. SteinUniversity of Connecticut

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SLIDE 61

Equiprobable Spaces

Definition (Equiprobable Space)

A probability space P is called an equiprobable space if each sample point has the same probability. If an equiprobabe space P has n points, we’ll write |P| = n, then p(x) = 1 n∀x ∈ P.

Alan H. SteinUniversity of Connecticut

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SLIDE 62

Equiprobable Spaces

Definition (Equiprobable Space)

A probability space P is called an equiprobable space if each sample point has the same probability. If an equiprobabe space P has n points, we’ll write |P| = n, then p(x) = 1 n∀x ∈ P. In an equiprobable space, if we have an event E ⊂ P, then p(E) = |E| n .

Alan H. SteinUniversity of Connecticut

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SLIDE 63

Equiprobable Spaces

Definition (Equiprobable Space)

A probability space P is called an equiprobable space if each sample point has the same probability. If an equiprobabe space P has n points, we’ll write |P| = n, then p(x) = 1 n∀x ∈ P. In an equiprobable space, if we have an event E ⊂ P, then p(E) = |E| n . It’s obviously important to be able to count the number of sample points comprising events.

Alan H. SteinUniversity of Connecticut

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SLIDE 64

Equiprobable Spaces

Definition (Equiprobable Space)

A probability space P is called an equiprobable space if each sample point has the same probability. If an equiprobabe space P has n points, we’ll write |P| = n, then p(x) = 1 n∀x ∈ P. In an equiprobable space, if we have an event E ⊂ P, then p(E) = |E| n . It’s obviously important to be able to count the number of sample points comprising events. The science of counting the size of a set is called combinatorics.

Alan H. SteinUniversity of Connecticut