Question

Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y. (Enter your answers in set notation.)

Answer 1

The outcomes corresponding to the five smallest possible values of Y are given by this set

{(S, S, S), (F,S,S,S), (F,F,S,S,S), (S,F,S,S,S), (F,F,F,S,S,S), (F,S,F,S,S,S), (S,F,F,S,S,S), (S,S,F,S,S,S), (F,F,F,F,S,S,S), (F,F,S,F,S,S,S), (F,S,F,F,S,S,S), (F,S,S,F,S,S,S), (S,F,F,F,S,S,S), (S,F,S,F,S,S,S), (S,S,F,F,S,S,S)}

Explanation:

You need to end with 3 success (the experiment ends there). The smallest possible value Y can take is 3, and it is given by

(S, S, S)

For Y = 4 we have only 1 possibility, the first trial is a failure and the following 3 ones are success

(F, S, S, S)

For Y = 5 we have

(F, F, S, S, S) and

(S,F,S,S,S)

For Y = 6

(F, F, F, S, S, S)

(F,S,F,S,S,S)

(S,F,F,S,S,S)

(S,S,F,S,S,S)

And lastly, for Y = 7, we have

(F,F,F,F,S,S,S)

(F,F,S,F,S,S,S)

(F,S,F,F,S,S,S)

(F,S,S,F,S,S,S)

(S,F,F,F,S,S,S)

(S,F,S,F,S,S,S)

(S,S,F,F,S,S,S)

Therefore, the total amount of possible outcomes corresponding to the five smallest possible values is

{(S, S, S), (F,S,S,S), (F,F,S,S,S), (S,F,S,S,S), (F,F,F,S,S,S), (F,S,F,S,S,S), (S,F,F,S,S,S), (S,S,F,S,S,S), (F,F,F,F,S,S,S), (F,F,S,F,S,S,S), (F,S,F,F,S,S,S), (F,S,S,F,S,S,S), (S,F,F,F,S,S,S), (S,F,S,F,S,S,S), (S,S,F,F,S,S,S)}