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(b) There are two villages. In the …rst village, each couple is allowed to have one child at most. In the other one there is no such a constraint. Suppose all parents want boys, so that in the second village each couple will keep trying until they get a boy. In the end, will the Boy/Girl ratios in the two villages be extremely di¤erent? Reason your answer. Answer: Intuitively, if each child independently has probability to be a boy and 1 probability to be a girl, then no matter how many children a couple already have, the probability of the next child being a boy is still : Comparing the two villages, although the numbers of children (per family) may be very di¤erent, the probabilities of each child being a boy are both . Thus the percentage of boys is approximately in each village. The boy/girl ratio should be approximately = (1 ): Formally, let random variable X be the number of girls in a family in the second village. Then under conventional assumptions, X has a geometric distribution (a special negative binomial distribution with r = 0). The p.d.f. of X is x f (x) = (1 ) and the expectation of X is E (X)
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Then Xi are independent and have the same distribution as X: Since the number of boys per family in the second village is 1; the boy/girl ratio of second village is Pn 1 1 Pni=1 = Pn Xi i=1 i=1 Xi n Using the law of large numbers (which we’ learn soon), the ratio converges to ll
1
almost surely.
(c) 11 cards, one letter on each, compose the word "probability". Now randomly pick 7 cards one by one and without replacement. What is the probability that the 7 letters on the 7 cards sequentially form the word "ability"? What is the probability that the 7 cards can be used (by reordering) to form the word "ability". Answer: (i) In total there are "ability", there are
11! (11 7)!
possible ordered sequences of the 7 cards, and for the them to be 1 2 2 1 (2 1) 1 1 = 4
possible cases, thus the probability is
4 4!
Then Xi are independent and have the same distribution as X: Since the number of boys per family in the second village is 1; the boy/girl ratio of second village is Pn 1 1 Pni=1 = Pn Xi i=1 i=1 Xi n Using the law of large numbers (which we’ learn soon), the ratio converges to ll
1
almost surely.
(c) 11 cards, one letter on each, compose the word "probability". Now randomly pick 7 cards one by one and without replacement. What is the probability that the 7 letters on the 7 cards sequentially form the word "ability"? What is the probability that the 7 cards can be used (by reordering) to form the word "ability". Answer: (i) In total there are "ability", there are
11! (11 7)!
possible ordered sequences of the 7 cards, and for the them to be 1 2 2 1 (2 1) 1 1 = 4
possible cases, thus the probability is
4 4!