Question

Problem 1. Assume an experiment with two outcomes success (S) or failure (F) is performed indefinitely. The outcomes of the experiments are assumed to be independent of each other. Assume the chance of success (S) in each trial is p. (a) Find the probability that the first S is observed at the 7th experiment. (b) Find a general formula for the probability that the first S is observed at the kth experiment. (c) What is the probability that the first S is observed at or after 7th experiment. (d) Find a general formula for the probability that the first S is observed at or after kth experiment.

Answer 1

Let X_i be the event that a S occurs at i^th experiment and thus X_i^c,  the event that a F occurs at i^th experiment. i=1,2,3,......

So, P[ X_i ] = p ; for all i = 1,2,...

and P[X_i^c] = 1- p = q , say ;for all i ;

a) Now, P[ First success occurs at 7th exp] = P₇ =  

[As Xi 's are independent]

  

b) General case P[First success occurs at k th exp]= P_k =

= q^(k-1).p

c) P[S is obsereved &th onwards ] = P₇+P₈+P₉+.......

=

=

=

= q⁶

d) Similarly P[First S occurs kth and onwards ] = q^(k-1) .

* This distribution is called geometric distribution.

If you have doubts, let me know below. thank you.