# Question: Here is another way to obtain a set of recursive

Here is another way to obtain a set of recursive equations for determining Pn, the probability that there is a string of k consecutive heads in a sequence of n flips of a fair coin that comes up heads with probability p:

(a) Argue that, for k < n, there will be a string of k consecutive heads if either

1. There is a string of k consecutive heads within the first n − 1 flips, or

2. There is no string of k consecutive heads within the first n − k − 1 flips, flip n – k is a tail, and flips n − k + 1, . . . , n are all heads.

(b) Using the preceding, relate Pn to Pn−1. Starting with Pk = pk, the recursion can be used to obtain Pk+1, then Pk+1, and so on, up to Pn.

(a) Argue that, for k < n, there will be a string of k consecutive heads if either

1. There is a string of k consecutive heads within the first n − 1 flips, or

2. There is no string of k consecutive heads within the first n − k − 1 flips, flip n – k is a tail, and flips n − k + 1, . . . , n are all heads.

(b) Using the preceding, relate Pn to Pn−1. Starting with Pk = pk, the recursion can be used to obtain Pk+1, then Pk+1, and so on, up to Pn.

**View Solution:**## Answer to relevant Questions

If X is a geometric random variable, show analytically that P{X = n + k|X > n} = P{X = k} Using the interpretation of a geometric random variable, give a verbal argument as to why the preceding equation is true. An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let X denote the selection number of the first chosen ball that is blue. ...Trains headed for destination A arrive at the train station at 15-minute intervals starting at 7 A.M., whereas trains headed for destination B arrive at 15-minute intervals starting at 7:05 A.M. (a) If a certain passenger ...If 65 percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of 100 people will contain (a) At least 50 who are in favor of the ...(a) A fire station is to be located along a road of length A, A < ∞. If fires occur at points uniformly chosen on (0, A), where should the station be located so as to minimize the expected distance from the fire? That is, ...Post your question