3 The multiplication rule/miscellaneous counting problems

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1 Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint, what is P (A B)? What is P (A B)? 2 Permutations and Combinations 2. Roll a die ten times. What the probability of getting exactly four 6 s? (For example, this occurs if you roll ). 3. In tossing 4 fair dice, what s the probability of tossing at most one 3? 4. How many anagrams of MISSISSIPPI are there? 5. The price of a certain stock changes each day at random: it goes up $1 with probability 0.6 and down $1 with probability 0.4. Assuming that changes on different days are independent, (a) what s the probability that it will be back to its starting price after 10 days? (b) what s the probability that it will be up by at least $1 after 4 days? 6. What is the coefficient on x 4 y 6 in the product (2x + 3y) 10? 3 The multiplication rule/miscellaneous counting problems 7. A hand of poker consists of 5 cards randomly drawn from a deck of 52 cards. What s the probability of getting 4-of-a-kind? (There are 13 kinds and 4 suits. Each card has a suit and a kind for a total of 4x13=52 cards). 8. A 5-card hand is dealt from a well-shuffled deck of 52 playing cards. What is the probability that the hand contains at least one card from each of the four suits? 9. You are dealt 6 cards from a standard deck of 52 cards. What is the probability that you get a four-of-a-kind and two cards that don t match. For example, you could get 4 Kings, a 10 and a 3. But you couldn t get 4 Kings and two 10s. 10. A licence plate consists of 3 letters followed by 3 numbers (for example XTY438). If no letter can be used more than once how many licence plates are there? The numbers can be repeated. For example XTY004 is valid.

2 11. In a bridge game, each of 4 players gets 13 cards drawn at random from an ordinary deck of 52 cards. What is the probability that two players each have 2 aces? There are 4 aces in the deck. 12. A certain group of 20 people consists of 7 doctors, 3 lawyers and 10 bankers. They are all seated at random around a round table. (a) What is the probability that the 3 lawyers sit next to each other? (b) What is the probability that no doctor sits next to another doctor? 13. A school play has 4 distinct male roles and 5 distinct female roles. If 7 men and 8 women audition for the play, how many possible casts are there? What if Bob and Alice refuse to be in the play together? 4 Urn problems 14. Urn #1 contains 3 black and 4 red balls. Urn #2 contains 5 black and 2 red balls. A ball is chosen at random from urn #1. A ball is also chosen at random from urn #2. What s the probability the two balls have the same color? 15. An urn contains 100 balls: 25 red, 25 blue, 50 green. Select 12 balls at random from the urn. (a) Assume that the selection is done without replacement. Compute the probability that 3 red, 4 blue ball and 5 green balls are selected. (b) Assume the same question if the selection is done with replacement. (c) What is the probability that all 12 balls have the same color? (assume the selection is done without replacement, then answer the same problem with replacement). (d) An urn contains 10 red balls and 5 blue balls. You grab 4 balls at random (without replacement). What is the probability that you grabbed 2 red balls and 2 black balls? 5 Inclusion/Exclusion and union problems 16. Suppose you draw 8 cards out of a regular deck of 52 cards. What is the probability that you will have at least 6 cards of the same suit? 17. In a certain city, 50% of the people speak Spanish, 45% speak English, 40% speak French, 15% speak Spanish and English, 15% speak English and French, 10% speak Spanish and French. If everyone speaks at least one of these three languages then what percentage of people speak all three? 18. Suppose you are dealt 5 cards at random from an ordinary deck of 52 cards. What s the probability of getting exactly four face cards? A face card is a king, queen or jack. There are 12 face cards in the deck.

3 19. Suppose that E 1, E 2, E 3, E 4 are four events such that P (E i ) = 0.4 (for all i), P (E i E j ) = 0.3 (for all i j), P (E i E j E k ) = 0.1 if i, j, k are distinct and P (E 1 E 2 E 3 E 4 ) = 0. What is P (E 1 E 2 E 3 E 4 )? 20. Suppose you draw 9 cards out of a regular deck of 52 cards. What is the probability that you will have at least 7 cards of one suit? For example, you could get 7 hearts and one diamond or 6 spades, 1 club and 1 heart. (There are 4 suits: hearts, diamonds, clubs and spades. There are 13 cards of each suit.) 21. Suppose you draw 8 cards out a regular deck of 52 cards. What is the probability that you get at least one 4-of-a-kind? (For example, this event occurs if you draw 4 Jacks and 4 Kings or if you draw 4 Jacks, a 2, a 3, a 4 and a 5.) Hint: Let E i be the event that you get 4 cards of kind i (i {1,..., 13} since there are 13 kinds). Compute P ( 13 i=1e i ) using inclusion-exclusion. 6 Conditional expectation 22. There are 3 cards in a hat. One is black on both sides, one is red on both sides and one is black on one side and red on the other side. A card chosen at random and placed (at random) on a table so that only one side is showing. Given that the side showing is red, what is the probability that the other side is also red? 23. Consider the following game. Roll a 6-sided die. If it lands on six, you win outright. If it lands on one, you lose. If it lands on 2, 3, 4, or 5 then you roll the die an additional 2, 3, 4, or 5 times respectively. If it lands on a 6 at least one time, then you win. Otherwise you lose. (a) What are your chances of losing given that you rolled a 3 on the first roll? (b) What are your chances of winning? 7 Recursive conditional probability problems 24. Alice and Bob are playing a game in which they take turns flipping a coin. The coin lands on heads with probability 1/3. Alice goes first. Alice wins if she flips heads before Bob flips tails. What s the probability that Alice wins? 8 Bayes 25. There are 2 coins in a box. The first lands on heads with probability 1/2. The second lands on heads with probability 1/3. You choose a coin at random (each possibility being equally likely) and flip it once. It comes up heads. (a) What is the probability that it was the first coin? (b) What is the probability that it will land on heads on the next flip?

4 26. The police hold 10 people for questioning in burglary. They know there are 2 burglars among the 10 people. So they give each person a lie detector test. The test is accurate with probability 0.9. This means that if the test is administered to a guilty person, it will say they are guilty with probability 0.9. If it is administered to an innocent person, it says they are innocent with probability 0.9. Suppose that a person, selected at random from amongst the 10, takes the test and it says guilty. What is the probability that he or she truly is guilty? 27. There are 3 coins in a box; one lands on heads with probability 1 (it has heads on both sides), one lands on heads with probability 1/2 and one lands one heads with probability 1/3. You choose a coin at random (each possibility being equally likely) and flip it 3 times. (a) What is the probability that it lands on heads all 3 times? (b) What is the probability that you chose the double-headed coin AND it landed on heads all three times? (c) Given that it lands on heads all three times, what is the probability that you chose the double-headed coin? (d) Given that it lands on heads all three times, what is the probability that it will land on heads on the next flip? 28. Suppose that there is a test for a certain disease with the following properties. If the subject has the disease the test results are positive with probability 0.9 and negative with probability 0.1. If the subject does not have the disease the rest results are positive with probability 0.1 and negative with probability 0.9. Assume 20% of the population has the disease. What is the probability that a randomly chosen person has the disease given that she tests positive? 29. There are 3 urns. Urn #1 contains 3 red balls and 4 blue balls. Urn #2 contains 4 red balls and 5 blue balls. Urn #3 contains 5 red balls and 6 red balls. An urn is chosen at random and a ball is selected from the urn. (a) What is the probability that a red ball is selected? (b) Given that a red ball is selected, what is the probability that Urn #1 was chosen? (c) Given that a red ball is selected, what is the probability that a second ball selected at random from the same urn (without replacement) is red? (d) There are two coins in a box. One is fair and the other is 2-headed. (The fair coin lands on heads 50% of the time). You choose a coin at random and flip it 10 times. It comes up heads each time. What s the probability that it is the 2-headed coin? 9 Independence 30. A coin lands on heads with probability 0.1. It is flipped 100 times. What is the probability that it lands on heads exactly 50 out of the 100 times?

5 31. A coin is tossed twice. Consider the events: A = heads on the first toss. B = heads on the second toss. C = the two tosses come out the same. (a) Are A, B independent? (b) Are B, C independent? (c) Are A, C independent? (d) Are A, B, C jointly independent? Justify your answer. 32. Suppose P (A) = 0.2, P (B) = 0.3. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint, what is P (A B)? What is P (A B)? 33. The price of a certain stock changes each day at random: it goes up $1 with probability 0.6 and down $1 with probability 0.4. Assuming that changes on different days are independent, what s the probability that it will be back to its starting price after 10 days?