"A first Course in probability"-chaper4-discrete random variable-negative two-term distribution

Based on our most familiar discrete distribution-two-item distributions, we can derive a number of other distribution columns, and for the previously introduced geometric distributions, we have given them the implication that the probability of success of an event is p, and the probability of the exact success of the N independent repetition experiment. Along the meaning of this layer, we have 1 times programmed R Times, then we get the so-called negative two distribution. A random variable with a negative two-item distribution is x, the probability of success for an independent event is p, and the probability of a successful R-time in an N-repeat independent experiment is:

Compared with two distributions, we can see that the negative two distributions emphasize the "just" success R times in n repetition experiments, that is to say that the nth experiment happens to be the first R successful experiment.

We have a question for example-the question of Barnabas matches.

Q: A smoking mathematician always carries two boxes of matches with him, one box in the left pocket and one in the right pocket. Every time he needs a match, he takes a match out of a matchbox in any pocket and now has n matches in each of the two boxes of matches, so when he first discovers that one of the boxes is empty, how likely is the other box to have a match of k?

Analysis: First of all we need to discuss a point is that the match is located in which pocket of the Matchbox is empty, obviously the left is the right symmetry, we analyze a situation, the square can be.

Assuming that the left pocket is empty, then the last step of the process is obviously to take a match from the right pocket when the mathematician first 2n-k the match, which is one of the last matches he took from his left pocket, and we can default the rational mathematician will not go to the matchbox with the left pocket, So we can associate it with a negative two-item distribution: In the 2n-k repeat experiment, there is a probability that n times out of the left pocket.

That

Of course, the final result of this problem should be squared as above probability.

"A first Course in probability"-chaper4-discrete random variable-negative two-term distribution