AC Geo/Adv AlgU4 Worksheet 4Name____________________________ FINDING PROBABILITIES FOR THE INTERSECTION OF EVENTS:P(AB) In most of the probability problems we have encountered so far, the intersection of the events we are interested in has been stated in the problem.We will now look at problems where you will be asked to find this intersection. There will be two situations for finding P(AB): 1) A and B are independent 2) A and B are dependent Multiplication Rule for Independent Events: P(AB)=P(A) P(B)(This rule also applies to multiple independent events) Ex1You toss a coin and roll a die.What is the probability you toss heads and roll a 3? P(H3) Ex2A bag contains 7 red, 4 green , and 9 yellow marbles.You draw one marble, replace it, and draw another.What is the chance that you drew a green first, then a yellow?What is the chance you drew red both times? P(GY)P(RR) If you draw three marbles, with replacement, what is the probability all three are yellow? Why is the replace it, important for this problem? Ex3You are given a 5 question multiple choice pop quiz.You know nothing about the subject, so you randomly guess on each question.There are four answer choices (a,b,c,d) for each question.What is the probability you get all five questions correct?What is the probability you get all five questions incorrect? General Multiplication Rule for Dependent Events: P(AB)=P(A)P(BA) orP(AB)=P(B)P(AB) Ex4 Consider the same bag of marbles from example 2.This time you do not replace the first marble you draw.What is the probability you draw a green, then a yellow?What is the probability you draw two reds? P(GY)=P(G)P(YG)P(RR)=P(first R)P(second Rfirst R) Ex5You draw two cards from a standard 52 card deck without replacement.What is the probability you draw a)A King and an ace b)Two Kings c)Two hearts d)A face card and an ace AssignmentU4 WS 4 In Exercises 1-2, determine whether the events are independent. 1. A single die is rolled and then rolled a second time. 2. Numbered balls are pulled from a bin one-by-one to determine the winning lottery numbers. Exercises 3-6, compute the conditional probabilities andP A B P B A ;

3. 0.7, 0.4, 0.25 P A P B P A B P(A B) = ) () (B PB and A P

4. 0.45, 0.8, 0.3 P A P B P A B 5. 0.61, 0.18, 0.07 P A P B P A B 6. 0.2, 0.5, 0.2 P A P B P A B In Exercises 7-8, determine whether the events are independent. 7. Numbers are written on slips of paper in a hat; one person pulls out a slip of paper without replacing it, then a second person pulls out a slip of paper. 8. In order to determine who goes first in a game, one person picks a number between 1 and 10, then a second person picks a number between 1 and 10. 9. A pair of dice are tossed. Find the probability that the sum on the two dice is 8, given that the sum is even. 10. A pair of dice are tossed. Find the probability that the sum on the two dice is 12, given that doubles are rolled. 11. A pair of dice are tossed. What is the probability that doubles are rolled, given that the sum on the two dice is less than 7? 12. A pair of dice are tossed. What is the probability that the sum on the two dice is 8, given that the sum is more than 6? 13. What is the probability of drawing two cards in succession (without replacement) from a standard deck and having them both be face cards? 14. Two cards are drawn from a standard deck without re- placement. Find the probability that both cards are hearts. 15. Two cards are drawn from a standard deck without replacement. What is the probability that the first card is a spade and the second card is red? 16. Two cards are drawn from a standard deck without replacement. What is the probability that the first card is a king and the second card is not? In Exercises 17-20, a snack-size bag of M&Ms candies is opened. Inside, there are 12 red candies, 12 blue, 7 green, 13 brown, 3 orange, and 10 yellow. Three candies are pulled from the bag in succession, without replacement. 17. Determine the probability that the first candy drawn is blue, the second is red, and the third is green. 18. Determine the probability that the first candy drawn is brown, the second is orange, and the third is yellow. 19. What is the probability that the first two candies drawn are green and the third is red? 20. What is the probability that the first candy drawn is orange, the second is blue, and the third is orange? In Exercises 21-24, three cards are dealt from a shuffled standard deck of playing cards. 21. Find the probability that the first card dealt is red, the second is black, and the third is red. 22. Find the probability that the first two cards dealt are clubs and the third is a spade. 23. What is the probability that the three cards dealt are, in order, an ace, a face card, and an 8? (A face card is a jack, queen, or king.) 24. What is the probability that the three cards dealt are, in order, a red card, a club, and another red card?