Probability Quiz Review Sections

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1 CP1 Math 2 Unit 9: Probability: Day 7/8 Topic Outline: Probability Quiz Review Sections Name A probability cannot exceed 1. We express probability as a fraction, decimal, or percent. Probabilities in situations where the outcomes are equally likely. o P(some event) =!"#$%&!"!"#$!%&'!"!"#!"!#$!"!#$!"#$%&!"!"#$!%&' If two events are independent (the result of A does not affect the result of B) o P(B A) = P(B) o P(A B) = P(A) o P(A and B) = P(A) P(B) Compound probabilities with or o If A and B are mutually exclusive (their sample spaces do not share outcomes), P(A or B) = P(A) + P(B) o If A and B are not mutually exclusive (their sample spaces share outcomes), P(A or B) = P(A) + P(B) P(A and B) Compound probabilities with and o If A and B are mutually exclusive, P(A and B) = 0 o For any two events A and B: P(A and B) = P(A) P(B A) o If events A and B are independent: P(A and B) = P(A) P(B) Note this is not actually different from this formula above, since P(B) = P(B A) for independent events Conditional probability: P(A B) = probability that event A happens given that event B happens Create and interpret tree diagrams to organize and find probabilities of multiple events Create and interpret Venn diagrams to organize and find probabilities of events A factorial problem is a problem that involves counting the number of ways that a set of things can be arranged in different orders. A permutation problem is a problem that involves counting the number of ways to select some things out of a group with an order. A combination problem is a problem that involves counting the number of ways that some things can be selected without choosing an order. Practice: Simplify all answers! 1. Lexington s theatre only has one ticket window. In how many ways can six people line up to purchase a ticket?

2 2. In my state s lottery, 48 balls are numbered from 1 to 48, and 6 are chosen. How many different sets of winning numbers are there? (In this lottery, the order in which the numbers are chosen does not matter). 3. The math club has 20 members. In how many ways can it select a president, a vice-president, and a treasurer if no member can hold more than one office? 4. If you flip a coin four times, what is the probability of getting 3 heads and 1 tails? Hint: Write out all the ways you can get 3 heads and 1 tails. 5. A board game has a spinner with the numbers 1 through 6 on it. All numbers are equally likely in a single spin. As you near the end of the game there are nine more spaces left to move. Find the probability that the sum of your two spins is at least 9. Hint: Write out all the ways you can get a sum of at least a. When drawing two cards from a standard deck of cards without replacement, what is the probability of drawing an odd numbered card (3, 5, 7 or 9), and then drawing a 10? b. When drawing two cards from a standard deck of cards with replacement, what is the probability of drawing an odd numbered card, and then drawing a 10?

3 7. A local gym wants to gather data on the levels of gym membership among men and woman in the community. Their research found that 52% of adults in the community are women, and 48% are men. 19% of the women have gym memberships, and 85% of men do not have a gym membership. a. Complete the following two-way table that summarizes the data. Round to the nearest hundredth. b. Calculate the following probabilities: I. P(G) IV. P(NG F) II. P(NG) V. P(M NG) III. P(F NG) VI. P(NG M) c. Are being male and having a gym membership independent? Show work and justify your answer with a written statement. 8. A jar contains 7 marbles total: 5 red marbles and 2 white marbles. If you draw two marbles without replacement, what is the probability that they are different?

4 9. Recently 400 students at Bridge School were surveyed about whether they ve seen two popular movies. Here are the results: 150 students saw Zootopia. 170 students saw Jungle Book. 160 students did not see either movie. a. Derman says, There must be an error in that data, because =480 which is more than the 400 students who were surveyed. What is the mistake in his reasoning? b. Make and label a Venn Diagram representing this situation. Don t forget the outer rectangle! c. On your Venn Diagram, shade the part that represents the students that did see Zootopia but did not see Jungle Book. d. If a surveyed student is chosen at random, what is the probability that the student did see Zootopia but did not see Jungle Book? e. If a surveyed student is chosen at random, what is the probability that the student saw both movies? f. If Z stands for seeing Zootopia and J stands for seeing Jungle Book, what is the probability P(Z or J)?

5 10. A drawer contains 5 red socks and 7 green socks. 2 socks are drawn without replacement. a. Draw a tree diagram to represent the conditional probabilities. In your diagram use red (R) and green (G) for the red and green socks. Label each branch with its conditional probability. b. What is the probability that the second sock is green given that the first sock is red? c. What is the probability that the second sock is green given that the first sock is green? = 6! = 720 Solutions 2.!"C! = 12,271,512 3.!"P! = P(3 heads and 1 tail) = 1/4 4 ways to get 3 heads and 1 tail HTHH, HHTH, THHH, HHHT 16 total outcomes when flipping a coin 4 times 5. P(sum of at least 9) =!"!" =!!" 10 ways to get a sum of at least 9 36 total outcomes when summing two spins 6. a.!"!"!!" =!"!!" b.!"!"!!" =!!"#

6 7. a. b. i. P(G) = 0.17 ii. P(NG) = 0.83 iii. P(F NG) = iv. P(NG F) = = 0.81 v. P(M NG) = = 0.49 vi. P(NG M) = = 0.85 (also given) c. No, they are not independent. Sample work: P(M)=0.48 but P(M G) = P(different colors) = P(white then red) + P(red then white) P(white marble followed by a red marble) =!! =!!!!" P(red marble followed by a white marble) =!! =!!!!" So, total probability of getting two different colored marbles =!"!" 9. a. He didn t consider that some students saw both movies, so they were counted in the 150 and the 170. b. If you used counts: 80 in overlap, 70 in Z only, 90 in J only, 160 outside both. If you used probabilities: 80/400 overlap, 70/400 Z only, 90/400 J only, 160/400 outside. c. Shade the part inside set Z but outside set J. d.!" =!!""!" e.!"!"" =!! f.!"#!"" =!! 10a. b. 7/11 c. 6/11