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 possile values of Y, and state which Y value is associated with each one.
Y Outcomes
3
4
5
6
7

Answer 1

Answer:

$(a)\ \{SSS\}$

$(b)\ \{FSSS\}$

$(c)\ \{FFSSS, SFSSS\}$

$(d)\ \{FFFSSS, SSFSSS, SFFSSS, FSFSSS\}$

$(e)\ \{FFFFSSS, FFSFSSS, FSSFSSS, SSFFSSS, SFSFSSS, SFFFSSS, FSFFSSS\}$

Step-by-step explanation:

Given

$S\to$ Success

$F\to$ Failure

$n \to 3$ Consecutive trials until S happens

$Y \to$ Number of trials

For every possible outcome, the last 3 must be SSS. So, we have:

$(a)\ Y = 3$

The only possibility here is: $\{SSS\}$

because 3 trials implies that all outcomes must be S.

$(b)\ Y = 4$

The only possibility here is:

$\{FSSS\}$

because 4 trials implies that the first outcome must be F

 

$(c)\ Y = 5$

The possibilities are:

$\{FFSSS, SFSSS\}$

because 5 trials implies that the first and the second outcomes must be FS or SF.

$(d)\ Y = 6$

The possibilities are:

$\{FFFSSS, SSFSSS, SFFSSS, FSFSSS\}$

because 6 trials implies that the first three outcomes are: FFF, SSF and SFF

$(e)\ Y = 7$

The possibilities are:

$\{FFFFSSS, FFSFSSS, FSSFSSS, SSFFSSS, SFSFSSS, SFFFSSS, FSFFSSS\}$

because 7 trials implies that the first three outcomes are: FFFF, FSSF and SSFF and FFSF (similar to (e))