Originally Given in December 1991
Note: This is only 6 of the 7 problems given in that exam. The remaining problem depended on a handwritten table that obviously didn't get stored on disk. This is the exam from F91, and the class was based on a different (terrible!) textbook. I did not do a very good job teaching that class (it was my first term here), and we didn't get as far as we should have. As a result, the final was somewhat more elementary than ours is likely to be.
Problem 1. Baseball
Cal Ripken plays shortstop for the Baltimore Orioles (my favorite baseball team). Suppose that every time he comes up to bat he has a 30% chance of getting a hit (independent of how he does in all his other times at bat). Suppose he comes up to bat 500 times in 1992.
a) What is the probability that he will finish the season with a batting average of .330 or better (i.e. that he will get 165 or more hits)? [For this part of the problem I want an EXACT answer (that is, some sort of formula), not a numerical approximation.]
a) This is a binomial distribution, with n=500 and p=.3. The probability of getting 165 or more hits is
b) Use the normal approximation to numerically approximate this probability.
b) , . 164.5 hits is 1.415 standard deviations above the mean, so our probability is approximately 1 - F(1.415) = 1-0.9214 = 0.0786, or 7.86%.
c) Using the normal approximation, estimate the probability that he finishes the season with a batting average of EXACTLY .330 (remember the continuity correction!).
c) The Z scores for 165.5 hits and 164.5 hits are 1.513 and 1.415, respectively, so our probability is F(1.513)-F(1.415) = 0.9349-0.9214=0.0135, or 1.35%.
Problem 2. Continuous distributions
The lifetime of a certain brand of light bulb is a random variable (call it X), with probability density function
a) Find the mean and the standard deviation of X.
a) (either make the substitution y=1+x or integrate by parts). . , .
b) What is the probability that a randomly chosen lightbulb lasts less than a year?
b) P(lives less than a year) =
c) Suppose we buy 5 lightbulbs. What is the probability that exactly 3 of these bulbs will burn out in the next year?
c) This is binomial, with n=5 and p=1/16 (see part b). The probability is , or about .025%
Problem 3. Conditional Probability
A system consists of 4 components (call them A, B, C, and D). On a given day, each component has a 50% probability of failing (independent of the others). If 3 or more components fail, the system breaks down.
a) What is the probability of the system breaking down today?
a) P(3 fail) + P(4 fail) = 4/16 + 1/16 = 5/16
b) Given that it broke down yesterday, what is the probability that component A failed?
b) The five (equally likely) possibilities are ABCD, ABC, ABD, ACD, and BCD. Four of the five have A failing, so the answer is 4/5.
Problem 4. Bridge
A bridge hand consists of 13 cards. For this problem you do not have to numerically evaluate your results. An answer like is just fine.
a) What is the probability that a bridge hand (13 cards) contains exactly 5 spades?
a) This is hypergeometric. .
b) What is the probability that a bridge hand contains exactly 5 spades AND exactly 5 hearts?
c) What is the probability that a bridge hand contains exactly 5 spades OR exactly 5 hearts (or both, of course)?
c) P(5S or 5H) = P(5S) + P(5H) - P(5S and 5H) = .
Problem 5. French Fish
Alain, Blaise, and Camille go fishing at Lac du Traute, somewhere in France. In France, catching fish is a Poisson process, with mean 0.15 fish/hour (per person). They spend 8 hours fishing on the lake.
a) What is the probability that Alain catches at least one fish in the first 4 hours?
a) For Alain for 4 hours, , so P(at least 1) .
b) What is the probability that Blaise catches exactly 2 fish in the day?
b) For 8 hours , so .
c) What is the probability that the 3 friends (they ARE friends) go home with a total of 3 fish?
For the three together, , so
Problem 6. Going Bananas
It's impossible to fill a banana crate to exactly a specified weight. If you aim for weight , you wind up with a normal distribution of weights with mean and standard deviation pounds.
a) What weight should the crate-fillers aim for in order to insure that 99% of the crates weigh at least 100 pounds?
a) To be 99% sure you need a 2.33 standard deviations cushion, so fill them to 111.65 pounds average.
b) Suppose the mean is set to be 110 pounds. A zookeeper buys 4 crates. What is the probability that he will get 430 pounds or more?
b) For the sum of 4 crates, the standard deviation is
pounds. Since the mean is 440, P(430 or more) = 1 - F(-1) = 0.8423.