8 Solved Questions Exam 1 - Statistics for the Life Sciences | MATH 20340, Exams of Mathematics

Download 8 Solved Questions Exam 1 - Statistics for the Life Sciences | MATH 20340 and more Exams Mathematics in PDF only on Docsity! Math 20340: Statistics for Life Sciences Fall 2008 Mid-semester Exam 1 Solutions 1. I roll a dice and look at the number that comes up (so there are six simple events, namely 1, 2, 3, 4, 5 and 6, and each occurs with probability 1/6). I am interested in the three events • A: the number I roll is 3 or smaller • B: the number I roll is 4 or larger • C: the number I roll is either 3 or 4 1. What is the probability of C? Solution: 1/3 2. Are the events A and B independent? Solution: No. P (A) = 1/2 and P (B) = 1/2 so P (A)P (B) = 1/4. But A ∩ B = ∅, so P (A ∩B) = 0. Since P (A ∩B) 6= P (A)P (B) the events are not independent 3. Are the events A and C independent? Solution: Yes. P (A) = 1/2 and P (C) = 1/3 so P (A)P (B) = 1/6. Also A ∩ C = {3}, so P (A ∩ C) = 1/6. Since P (A ∩ C) 6= P (A)P (C) (in other words, P (A|C) = P (A)) the events are independent 4. Write down a pair of events (from among A, B and C) that are mutually exclusive. Solution: A and B are mutually exclusive (A ∩B = ∅) 2. 60% of homeowners in St. Joseph County have fire insurance. Three homeowners are chosen at random, and x of them are found to have fire insurance. 1. What are the possible values for x? Solution: 0, 1, 2, 3 2. What are the probabilities of each of the possible values? Solution: p(0) = .43 = .064, p(1) = 3 ∗ .42 ∗ .6 = .288, p(2) = 3 ∗ .4 ∗ .62 = .432, p(3) = .63 = .216 3. What is the probability that at least two of the three are insured against fire? Solution: p(2) + p(3) = .648 1 3. A man takes either the bus or the subway to work each day. 70% of the time he takes the bus, and the rest of the time he takes the subway. When he takes the bus, he is late 10% of the time. When he takes the subway, he is late 5% of the time. 1. What is the probability that he takes the bus AND is late for work on a given day? Solution: P (B ∩ L) = P (B)P (L|B) = .7 ∗ .1 = .07 2. What is the probability that he is late for work on a given day? Solution: P (L) = P (B∩L)+P (S∩L) = .07+P (S)P (L|S) = .07+.3∗.05 = .07+.015 = .085 3. On one particular day, the man is late for work. What is the probability that he took the bus that day? Solution: P (B|L) = P (B ∩ L)/P (L) = .07/.085 = .823... 4. A student regularly visits Blue Chip Casino. 40% of the time that he visits, he gets past security and onto the playing floor. If he gets onto the playing floor, he rolls a dice, and if it comes up 2 or 5 he plays craps, but if it comes up 1, 3, 4 or 6 he plays poker. He is good at both, and always wins $200 at the craps table and $100 at the poker table. 1. Let x be the amount of money that the student wins visiting the casino. What are the possible values that x can have? Solution: 0, 100, 200 2. What are the probabilities of each of the possible values for x? Solution: p(0) = .6, P (100) = .4 ∗ (2/3) = .26..., P (200) = .4 ∗ (1/3) = .13... 3. What is the expected value of x? Solution: E(x) = 0 ∗ .6 + 100 ∗ .26... + 200 ∗ .13... = 53.33... 4. How much can the student afford to pay traveling to the casino so that his expected gain is zero? Solution: $53.33 2