Probability Theory and Statistics

1. Further generalize the Bayesian analysis of medical screening from last week. Let a rare disease D occur with probability P(D) = δ, with δ small. Now suppose that the test has different (but still small) error rates for positive and negative results: P(+|D) = (1 − ε+ ) and P(−|Dc) = (1 − ε−). Derive the probabilities P(Dc|+) and P(D|−). Interpret your results.
2. In a particular game, you keep rolling a pair of (six-sided) dice until the sum comes up either “7” or “11”; once this happens you stop (and someone else rolls). Let Ri be the event that you stop after roll number i. Calculate P(R1) and P(R2), and find a general formula for P(Rn).
3. In a five-card poker hand, what is the probability of getting “two pair,” with two cards of the same kind, two cards of a different kind, and one card of a third kind?
4. Derive the multinomial coefficient for the number of ways to divide n objects into three groups of n1, n2 and n3 objects, with n1 +n2 +n3 = n; second group.) Confirm that this reduces to the usual binomial coefficient when n3 = 0.
5. (Mendelian genetics.) Pea plants have green seeds or yellow seeds. The color is controlled by two genes, one from each parent plant, with yellow (y) being dominant over green (g). A gg plant has green seeds, yy has yellow, and the hybrids yg and gy have yellow seeds. In the wild, the four types {yy,yg,gy,gg} are equally likely.
   1. (a) Two wild plants with yellow seeds are randomly selected and bred to give a first generation of plants. What is the probability that a first-generation plant has yellow seeds? What is the probability that it has yy genes?
   2. (b) Two first-generation plants with yellow seeds are randomly selected and bred to give second-generation plants. What is the probability that a second-generation plant has yellow seeds? What is the probability that it has yy genes?
6. A Starburst candy package has 12 pieces, three each of four flavors: berry, lemon, orange and cherry. The pieces are arranged randomly in the pack. You draw the first four from the pack. (a) What is the probability that all four are different flavors?
   (b) What is the probability that your four pieces have exactly two each of two flavors? (c) What is the probability that your four pieces include all the flavors but lemon?

n n! n,n,n =n!n!n!

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(Hint: first divide the n objects into groups of n1 and n − n1 objects, and then subdivide the

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