Midterm 2 Practice Questions for First Year Interest Group Seminar | N 1, Exams of Health sciences

Download Midterm 2 Practice Questions for First Year Interest Group Seminar | N 1 and more Exams Health sciences in PDF only on Docsity! Math 362K Probability Fall 2007 Instructor: Geir Helleloid Midterm 2 Practice Questions 1. Practice Midterm 1 (a) What is the probability that the first card in a deck of 52 cards is a spade, the second card is a spade, and the third card is a heart? (b) If A flips n + 1 coins and B flips n fair coins, find the probability that A gets more heads than B. Hint: Condition on which player has more heads after each has flipped n coins. (There are three possibilities.) (c) Suppose that 5 percent of men and .25 percent of women are colorblind. A colorblind person is chosen at random. What is the probability of this person being male? Assume that there are equal numbers of men and women. (d) I invite 10 people to dinner. I estimate that each person will be able to come to dinner with probability 0.6 (independent of the other invitees). What is the probability that at least 8 people will be able to come? (e) The average number of major hurricanes near the United States every year is 2.6. Estimate the probability that there are more than 4 major hurricanes in one year. (f) The probability density function f(x) of a continuous random variable Q is f(x) = { 0 : x < 0 4e−cx : x ≥ 0 for some constant c. i. Find the value of c. ii. Find the cumulative distribution function F (b) of Q. iii. Find the probability that Q is between 1 and 3. iv. Find the expected value of Q. v. If Q represents the lifetime of a lightbulb in years, find the probability that of three such lightbulbs, exactly two last more than two years. 2. Practice Midterm 2 (a) What is the probability that two cards drawn from a 52-card deck form a pair (that is, have the same rank) given that they are both black cards (that is, each card is either a spade or a club)? (b) There are two local factories that produce radios. Each radio produced at factor A is defective with probability .05, whereas each one produced at factory B is defective with probability .01. Suppose you purchase two radios that were pro- duced at the same factory, which is equally likely to have been either factory A or factory B. If the first radio that you check is defective, what is the conditional probability that the other one is also defective? (c) English and American spellings are “rigour” and “rigor”, respectively. A man staying at a Parisian hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40 percent of the English-speaking men at the hotel are English and 60 percent are Americans, what is the probability that the writer is an Englishman? (d) A and B flip coins. A starts and continues flipping until a tail occurs. At this point, B starts flipping and continues until there is a tail, then A takes over, and so on. The probability that A gets heads on a coin flip is .3 and the probability that B gets heads on a coin flip is .5. The winner of the game is the first one to get 2 heads in a row. What is the probability that A wins? (e) A local soccer team has 5 more games that it will play. If it wins its game this weekend, then it will play its final 4 games in the upper bracket of its league, and if it loses, then it will play its final games in the lower bracket. If it plays in the upper bracket, then it will independently win each of its games in this bracket with probability .4, and if it plays in the lower bracket, then it will independently wins each of its games with probability .7. If the probability that it wins its game this weekend is .6, what is the probability that it wins at least 3 of its final 4 games? (f) The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probability that there will be at most 1 such accident in the next month? (g) For some constant c, the random variable Q has probability density function f(x) = { cx4 : 0 < x < 2 0 : otherwise Find E[Q] and Var(Q). 3. Practice Midterm 3 (a) What is the probability that when three fair dice are thrown, one of the dice is showing a 5, given that the three numbers showing are all different? (b) A total of 48 percent of the women and 37 percent of the men that took a certain “quit smoking” class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of a year. If 62 percent of the original class were male, what percentage of those attending the party were women? What percentage of the original class attended the party? (c) Factories A, B, and C produce 50, 75, and 100 computers respectively. Computers from factory A have a 10 percent failure rate in the first year, those from factory B have a 20 percent failure rate, and those from factory C have a 25 percent failure rate. Given one of the 225 total computers, if it fails in the first year, what is the probability that it came from factory A? (d) A coin that when flipped comes up heads with probability p is flipped until either heads or tails has occurred twice. Find the expected number of flips.