Stat 20 Study Group Faculty: Professor Hank Ibser Study Group Leader: Larry Wang, larry@csrjjsmp.com Location: MW 1-2 201A Chavez , http://www.csrjjsmp.com/stat20.html Community through Academics and Leadership Worksheet #9: Probability 1. Consider two events, A and B. Suppose P(A) = 99% and P(B) = 44%. a. What is the chance that either A or B occurs? b. What is the chance that they both occur? c. How do your answers change if you know the events are independent? 2. In a class of 30, there are 18 male students and 12 female students. 10 each have A’s, B’s, and C’s. A student’s grade and sex are independent. a. How many female students have B’s? b. How many students are female or have B’s? c. Choose 2 students at random from the class without replacement. What is the probability that they are both male and have different grades? d. Choose 10 students at random from the class without replacement. What is the probability that they all have B’s? How can we write this concisely? e. Choose 10 students at random from the class, with replacement. What is the probability that 5 are male and 5 are female? f. Choose 10 students at random from the class, with replacement. What is the probability that 3 have A’s, 3 have B’s, and 4 have C’s? 3. You flip four fair independent coins, labeled 1 through 4. Find the probability that: a. You get more heads than tails. b. You get at least one head c. You get at least one head and at least one tail. d. You get more heads than tails and (you get at least one head and at least one tail). e. You get more heads than tails or (you get at least one head and at least one tail). f. You get more heads than tails given (you get at least one head and at least one tail). g. You get 1 head in the first two flips and you get 1 head in the last two flips 4. Gauss and Euler are playing poker. 5 cards are dealt to each player. A “Full House” is a specific poker hand that consists of a “two-of-a-kind” and a “three-of-a-kind”. One example would be KKKQQ, in any order. A “Two Pair” consists of 2 different pairs and a fifth non-matching card. An example of this would be “AAQQ3”, also in any order. a. What is the probability that Gauss gets a full house with 444JJ in that order? b. What is the probability that Gauss gets a full house with 444JJ in any order? c. What is the probability that Euler gets a full house with JJJ44 in any order? d. What is the probability that Euler gets a full house by first drawing 3 Jacks, then 2 of some other card? e. What is the probability that Euler gets a full house by first drawing 3 of any card, then 2 of some other card? f. What is the probability that Euler gets a full house? g. What is the probability that Euler gets a full house with JJJ44 if Gauss has… i. Three 4s and two Jacks? ii. Three Queens and two 3s? iii. Three Queens and two 4s? Stat 20 Study Group Faculty: Professor Hank Ibser Study Group Leader: Larry Wang, larry@csrjjsmp.com Location: MW 1-2 201A Chavez , http://www.csrjjsmp.com/stat20.html h. i. j. k. Community through Academics and Leadership What is the probability that both players get full houses? What is the probability that Gauss two Jacks, two Queens, and a 7 in any order? What is the probability that Gauss has two Jacks, two Queens, and any other card? What is the probability that Gauss gets a two pair? 5. You are playing a lottery where each player chooses a number from 0-99999, and 5 numbers 0-9 are drawn. If any rearrangement of drawn numbers matches a player’s number, he wins. What is the probability that a player will win if he chooses the number: a. 99999? c. 94704? b. 12345? d. 66778? e. What kind of number should you choose to maximize your chance to win? 6. Consider the sentence "the simians are jubilant." Neglecting the period at the end, but including the spaces, there are 24 characters in this sentence. Suppose we put 1,000 monkeys in front of special typewriters that have only lowercase letters and a spacebar—no numbers, punctuation marks, or special characters, so there are 27 keys in all. Every time a monkey types 24 characters, we change the paper in the typewriter. Assume that monkeys type independently of each other, and that they pick which character to type independently each time and with equal probability of striking each of the 27 keys. Each monkey types 24 characters per minute for 8 hours a day, 365.25 days a year, for 20 years. Assume that it does not take any time to change the paper in the typewriter. What is the chance that at least one of the monkeys types the sentence? You can use the approximation 1 − (1−x)n is approximately n×x. (This is valid for very small x and essentially means we ignore the probability the sentence is typed multiple times) 7. Cards are dealt without replacement from a standard deck of cards until a heart is dealt. a. What is the longest you might have to wait? b. What is the probability that exactly 5 cards are required? c. What is the probability that 5 or fewer cards are required? d. Given that exactly 5 cards are required, what is the probability that 3 spades were dealt? 8. Keith sets up a maze for his mouse Ralph. Whenever Ralph reaches an intersection, he randomly chooses a path to go down other than the one he just came out of. If he reaches a dead end, he turns back the way he came. Given each maze, what is the probability Ralph reaches the food? What is the probability he comes back out the entrance?