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Chapter 4: Q.4.73 (page 168)
Suppose in Problem 4.72 that the two teams are evenly matched and each has probability 1 2 of winning each game. Find the expected number of games played.
In the given information the answer is
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Here is another way to obtain a set of recursive equations for determining , the probability that there is a string of consecutive heads in a sequence of flips of a fair coin that comes up heads with probability :
(a) Argue that for , there will be a string of consecutive heads if either
1. there is a string of consecutive heads within the first flips, or
2. there is no string of consecutive heads within the first flips, flip is a tail, and flips are all heads.
(b) Using the preceding, relate . Starting with , the recursion can be used to obtain , then, and so on, up to .
Show that is a Poisson random variable with parameter , then
Now use this result to compute .
One of the numbers through is randomly chosen. You are to try to guess the number chosen by asking questions with 鈥測es-no鈥� answers. Compute the expected number of questions you will need to ask in each of the following two cases:
(a) Your ith question is to be 鈥淚s it i?鈥� i = . (b) With each question, you try to eliminate one-half of the remaining numbers, as nearly as possible.
If the distribution function of is given by
calculate the probability mass function of .
A family has n children with probability where
(a) What proportion of families has no children?
(b) If each child is equally likely to be a boy or a girl (independently of each other), what proportion of families consists of k boys (and any number of girls)?
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