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Warm Up Chapter 6.2 Quiz Today 1. What is the probability of choosing an Ace, with out returning it, and then a King from a deck of cards? Assume the deck has 52 cards. 2. Redo #1 but this time return the card. AP Statistics, Section 6.3, Part 1 2 Section 6.3.1 Probability Models AP Statistics Venn Diagrams: Disjoint Events S A B S = (sample space=all possible outcomes.) Not the same as Independent: Independent events must be able to occur at the same time. If one happens, it has no influence on the other whatsoever. The occurrence of one provides no information about the other. 4 Independent vs Disjoint The occurrence of one provides no information about the other. Ex: You place a bet on a card being red. I peak and tell you it’s an ace. Does that help you? Before you knew this, the probability the card is red was 26/52 = 1/2. Knowing it’s an ace, the probability it’s red is 2/4 = 1/2. No help whatsoever – the probability has not changed. These two events ARE independent (and not disjoint). P(red | ace) = P(red) — that’s the very definition of independence: the occurrence of “ace” has no effect on the probability of “red”. 5 Venn Diagrams: Disjoint Events P( A or B) P( A) P( B) S A B AP Statistics, Section 6.3, Part 1 6 Venn Diagrams: Non-disjoint Events P( A or B) P( A) P( B) P( A and B) S B A A and B AP Statistics, Section 6.3, Part 1 7 Venn Diagrams: Non-disjoint Events P( A B) P( A) P( B) P( A B) S B A A and B AP Statistics, Section 6.3, Part 1 8 Example Deborah and Matthew are awaiting the decision about a promotion. Deborah guesses her probability of her getting a promotion at .7 and Matthew’s probability at .5. AP Statistics, Section 6.3, Part 1 9 Example Deborah and Matthew are awaiting the decision about a promotion. Deborah guesses her probability of her getting a promotion at .7 and Matthew’s probability at .5. D .7 AP Statistics, Section 6.3, Part 1 M .5 10 Example Since there is not enough information to do the problem, let’s add information. Deborah thinks the probability of both getting promoted is .3 .1 D .4 .7 AP Statistics, Section 6.3, Part 1 D and M .3 M .5 .2 11 Example .1 What’s the probability of only Deborah getting promoted P(D-M)? P(M-D)? P(Dc)? P(Mc)? P(Dc and Mc)? D .4 AP Statistics, Section 6.3, Part 1 D and M .3 M .2 12 Different Look Matthew Promoted Promoted Deborah Not Promoted .3 Total .7 Not Promoted Total .5 AP Statistics, Section 6.3, Part 1 13 Different Look Matthew Promoted Promoted Deborah Not Promoted Total .3 .7 .5 1.0 Not Promoted Total AP Statistics, Section 6.3, Part 1 14 Different Look Matthew Deborah Promoted Not Promoted Total Promoted .3 .4 .7 Not Promoted .2 Total .5 AP Statistics, Section 6.3, Part 1 1.0 15 Different Look Matthew Deborah Promoted Not Promoted Total Promoted .3 .4 .7 Not Promoted .2 Total .5 AP Statistics, Section 6.3, Part 1 .3 .5 1.0 16 Different Look Matthew Deborah Promoted Not Promoted Total Promoted .3 .4 .7 Not Promoted .2 .1 .3 Total .5 .5 1.0 AP Statistics, Section 6.3, Part 1 17 3 Events Preference in Pizza Toppings for 25% like pepperoni 20% like combination 30% like cheese Whole circle needs to kids. add up to 25% Pepperoni Also 10% like both pepperoni and cheese Combination 5% like all three 12% like both combination and cheese 10% like pepperoni only Whole circle needs to add up to 20% AP Statistics, Section 6.3, Part 1 Cheese Whole circle needs to add up to 30% 18 3 Events Preference in Pizza Toppings for kids. 25% like pepperoni 52% 20% like combination 30%like cheese Pepperoni 10% Also Sample Space Mustpepperoni 10% like both Add Up to cheese100% 5% and like all three 12% like both combination and cheese 10% like pepperoni only Combination 3% AP Statistics, Section 6.3, Part 1 5% 5% 5% 7% Cheese 13% 19 3 Events Preference in Pizza Toppings for kids. What percent of kids like only Combination? 52% Pepperoni 10% 3% What percent kids like none of the three? 52% What 5% percent like Cheese but not combination? 5% 5% Combination 3% 7% Cheese 13% 18% AP Statistics, Section 6.3, Part 1 20 Using a Venn Diagram The probability of event A happening is .64 .28 The Probability of event B happening is .20 The probability of both events occurring is .12. What is the probability of each event. A and B =.12 A and Bᶜ =.52 =.08 Aᶜ and B Aᶜ and Bᶜ =.28 A .52 B .12 .08 AP Statistics, Section 6.3, Part 1 21 Assignment Exercises: 6.46-6.53, all AP Statistics, Section 6.3, Part 1 22