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Assignment Physics #12
Spring 2006
The topics this week are Probability and Statistics. Readings: Riley, Hobson and Bence - Chapters 26, 27
Boccio - 13_ProbStat Problems: 26.03 Give advice? EP-34 There are 3 coins in a box. One is two-headed. One is a fair coin. The third coin is biased and comes up heads 75% of the time. A coin is selected at random, flipped, and it shows heads. What is the probability that it was the
two-headed coin? EP-35 An urn contains 5 white and 4 black balls. A sample of 4 balls is picked from the urn.
(a) If the sample was picked without replacement, what is the probability that the sample contains 2 white and 2 black balls? (b) If the sample was picked with replacement, what is the probability that the first 2 balls picked are white and the last 2 balls picked are black? (c) Your friend picks 4 balls without replacement and tells you that at least 2 of them are white. What is the probability that 2 of them are black?
EP-36 After your yearly checkup, the doctor has bad news and good news. The bad news is that you tested positive for a serious disease, and that the test is 99% accurate(i.e., the probability of testing positive given that you have the disease is 0.99, as is the probability of testing negative if you don't have the disease). The good news is that this is a rare disease, striking only 1 in 10000 people. Why is it good news that the disease is rare? What are the chances that you actually have the disease?
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EP-37 There are 3 envelopes that look alike. One contains two $1 bills, one contains two $100 bills, and one contains one $1 bill and one $100 bill. You are permitted to choose one envelope. Then you are asked to remove one bill from the envelope without looking at the other bill. Suppose that a $100 bill comes out. What is the probability that the other bill is $100. EP-38 Poisson Distribution
(a) A roll of fabric has an average of 2 defects per 100 square yards. If seven 10 square yard specimens are chosen randomly, what is the probability that the total number of defects will be 4? (b) Assume that typographical errors committed by a typesetter occur completely at random. Suppose that a book of 600 pages contains 600 such errors. Calculate the probability
(1) that a page contains no errors
(2) that a page contains at least 3 errors
EP-39 In the game of Russian roulette (this is a theoretical problem not meant to be tried experimentally), one inserts a single bullet into the drum of a revolver, leaving the other five chambers of the drum empty. One then spins the drum, aims at one's head, and pulls the trigger. (a) What is the probability of still being alive after playing the game N times? (b) What is the probability of surviving (N-1) turns in this game and then being shot the Nth time one pulls the trigger? (c) What is the mean number of times a player gets the opportunity of pulling the trigger in the macabre game?
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EP-40 A set of telephone lines is to be installed so as to connect town A to town B. The town A has 2000 telephones. If each of the telephone users of A were to be guaranteed instant access to make calls to B, 2000 telephone line would be needed. This would be rather extravagant. Suppose that during the busiest hour of the day each subscriber in A requires, on the average, a telephone connection to B for 2 minutes, and that these telephone calls are made at random. the Find minimum number M of telephone lines to B which must be installed so that only 1% of the callers of town A will fail to have immediate access to a telephone line to B (approximate the distribution by a Gaussian distribution). EP-41 Consider a set of ten urns. Nine of them contain three white chips and three red chips each. The tenth urn contains five white chips and one red chip. An urn is picked at random and a sample of size four is picked (without replacement) from that urn. Three of the four chips turn out to be white, and the fourth turns out to be red. What is the probability that the urn sampled was the one with the five white chips? EP-42 A multiple choice exam gives 5 choices per question. On 75% of the questions, you think you know the answer; on the other 25% of the questions, you just guess the answer. Unfortunately, when you think you know the answer, you are right only 80% of the time. (a) Find the probability of getting an arbitrary question right. (b) If you do get a question right, what is the probability that it was a lucky guess? EP-43 The number X of electrons counted by a receiver in an optical communication system is a Poisson random variable with rate 1 when the signal is present and with rate 0 < 1 when the signal is absent. Suppose that a signal is present with probability p. (a) Find P(signal present|X=k) and P(signal absent|X=k)
Page 3
(b) The receiver uses the following decision rule: If P(signal present|X=k) > P(signal absent|X=k), decide signal present, otherwise decide signal absent. Show that this decision rule leads to the following threshold rule: If X > T, decide signal present, otherwise, decide signal absent. (c) What is the probability of error for the above rule? EP-44 Chi-squared goodness of fit test A ch-squared test can be used to test the hypothesis that observed data follow a particular distribution. The test procedure consists of arranging the n observations in the sample into a frequency table with k classes. The chi-squared statistic is: ( O E )2 2 = E
where O = observed frequency and E = expected frequency. The number of degrees of freedom k - p - 1, where p is the number of parameters. Let us consider an example involving the Poisson distribution. Let X be the number of defects in printed circuit boards. A random sample of n = 60 boards is taken and the number of defects recorded. The results were as follows:
Number of Observed Defects Frequency ---------------------
0
32
1
15
2
9
3
4
Does the assumption of a Poisson distribution sewem appropriate for these data? (Chi tables below).
Page 4
EP-45 More chi-squared We want to determine if a coin is fair. In other words, are the odds of flipping the coin heads-up the same as tails-up. We collect data by flipping the coin.
(a) Suppose a coin is tossed 10 times and 7 heads are observed. Based on this experiment, is the coin fair? (b) Now the coin is tossed 100 times and 70 heads are observed. Is the coin fair?
(c) Now we consider a more realistic case. We flip the coin 200 times. The coin landed heads-up 108 times and tails-up 92 times. At first glance, we might suspect that the coin is biased because heads resulted more often than than tails. However, we have a more quantitative way to analyze our results, a chi-squared test. Is this coin fair?
(Chi tables below).
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