Intersections and Unions of Events CS 70, Summer 2019 Lecture 17, - - PowerPoint PPT Presentation

Intersections and Unions of Events

CS 70, Summer 2019 Lecture 17, 7/23/19

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Last Time: Conditional Probability

I P[A|B]:restricting the sample space to B

P[A ∩ B] = P[A|B] P[B] = P[B|A] P[A]

I “Total probability rule:” probability by disjoint cases I “Bayes’ Rule”: definition of conditional probability +

total probability rule

I Lets you “flip” the conditioning 2 / 25

Computing Intersections

For any events A, B: P[A ∩ B] = What about any (three) events, A, B, C? P[A ∩ B ∩ C] =

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Computing Intersections: Chaining

For any events A1, A2, . . . , An: P " n \

i=1

Ai # = P[A1] · P[A2|A1] · P[A3|A1 ∩ A2] · . . . Proof: Details are in the notes. General Idea: Key Insight: Treat (A1 ∩ A2 ∩ . . . ∩ An−1) as one event, treat An alone as another. Very similar to...

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I draw 4 cards sequentially from a standard deck, without

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Ci = P[C1 ∩ C2 ∩ C3 ∩ C4] =

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(Modified from notes.) I am dealt 5 cards. What is the probability that all five cards are the same suit, and none of them are face cards? C1 = For 2 ≤ i ≤ 5: Ci = P[C1 ∩ C2 ∩ C3 ∩ C4 ∩ C5] =

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Independent Events

For any events A, B: P[A \ B] = P[A|B] · P[B] = P[B|A] · P[A] Two events A, B are independent if and only if: P[A \ B] = which is equivalent to both: P[A|B] = P[B|A] =

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Independent or Not?

I Flipping two fair coins:

A = flip 1 is heads, B = flip 2 is tails.

I Rolling one red die, one blue die:

A = sum is 3, B = red die is 1.

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Independent or Not?

I Rolling one red die, one blue die:

A = sum is 7, B = red die is 1.

I Throwing 3 labeled balls into 3 labeled bins:

A = Bin #1 is empty, B = Bin #2 is empty.

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Independent or Not?

A, B are generic events. The table shows probabilities of the intersections of the row and column. Event A Event A Event B P[A \ B] = 0.4 P[A \ B] = 0.3 Event B P[A \ B] = 0.2 P[A \ B] = 0.1 What is P[A]? What is P[B]? Are A and B independent?

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Mutual Independence

How do we generalize independence from two events A, B, to multiple events A1, A2, . . . , An? Definition (Mutual Independence, Ver. 1) A1, A2, . . . , An are mutually independent if: For every I ✓ {1, 2, . . . , n}, with |I| 2, P "\

i∈I

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Mutual Independence

Definition (Mutual Independence, Ver. 2) A1, A2, . . . , An are mutually independent if: For every choice of Bi 2 {Ai, Ai}: P[B1 \ B2 \ . . . \ Bn] =

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A Weaker Idea: Pairwise Independence

Definition (Pairwise Independence) A1, A2, . . . , An are pairwise independent if: For every i 6= j in {1, 2, . . . , n}: P[Ai \ Aj] = P[Ai] · P[Aj] Q: Does mutual imply pairwise? Q: Does pairwise imply mutual?

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Using (Mutual) Independence: Coin Flips

What is the probability that after n flips, we have k heads and (n k) tails?

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Using (Mutual) Independence: Dice Rolls

We roll n red dice and n blue dice. What is the probability that all the red dice are even, and all the blue dice are 5?

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Exercise

Break

Back by popular demand... Would you rather only use spoons (no forks) or only use forks (no spoons) for the rest of your life? A joke... Why was 6 afraid of 7? Because seven ate nine. Now, why was 7 afraid of 8?

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Unions of Events

Same exact story as in counting... P[A [ B] =

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Unions of Events

Same exact story as in counting... P[A [ B [ C] =

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Unions Example: Rolling 3 Die

I roll a red die, a blue die, and a green die. What is the probability that at least one of these happen? A) The red die’s number is 3, or 4 B) The blue die’s number is 5. C) The green die’s number is 1 or 6. P[A] = P[B] = P[C] =

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Example: Rolling 3 Die

Continued... P[A \ B] = P[A \ C] = P[B \ C] = P[A \ B \ C] = P[A [ B [ C] =

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Principle of Inclusion and Exclusion

Same exact story as in counting... For probability: Let A1, A2, . . . , An be events in our probability space. Denote {1, 2, . . . , n} by [n]. Then:

P " n [

i=1

Ai # = X

{i}✓[n]

P[Ai] X

{i,j}✓[n]

P[Ai \ Aj] + X

{i,j,k}✓[n]

P[Ai \ Aj \ Ak] . . . . . . + (1)n+1 P[A1 \ A2 \ . . . \ An]

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The Union Bound

Q: What is the maximum possible value of the following? P[A1 [ A2 [ . . . [ An] A: P[A1 [ A2 [ . . . [ An] is always upper bounded by

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Union Bound Example: Rolling 3 Die

I roll a red die, a blue die, and a green die. What is an easy upper bound on the probability that at least one of these happen? A) The red die’s number is 3 or 4. B) The blue die’s number is 5. C) The green die’s number is 1 or 6.

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Summary

I Computing event intersections = chaining conditional

probabilities

I Independent events = directly multiply probabilities I Mutual independence 6= pairwise independence

I Computing event unions = same exact strategy from

counting!

I Draw the Venn diagram for 2 events, 3 events I Principle of Inclusion-Exclusion for multiple events

I Union bound = worst case, the events are disjoint!

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Tips for Counting and Probability

I Don’t overthink it! Consider one thing at a time. I Label your events!! Be cognizant of whether or not

you are conditioning.

I If you have time, try a different strategy and see if it

gets you the same answer (e.g. cases vs. complement)

I Try small examples to sanity-check your strategy! I Practice, practice, practice!

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