Question

Six​ stand-up comics,​ A, B,​ C, D,​ E, and​ F, are to perform on a single evening at a comedy club. The order of performance is determined by random selection. Find the probability​ that: a. Comic D will perform first. b. Comic C will perform first and Comic E will perform fourth. c. The comedians will perform in the following​ order: D, C, E, A, B, F. d. Comic B or Comic F will perform third.

Answer 1

Probability of any event is defined by

`P(E)=(favourable case)/(Total cases)`

a)

Since any of the six comedians can perform first so the probability that comic D will perform first is given by
`(1)/(6)`

b)

The probability of event that comic C will perform first and comic E will perform fourth is given by

`P(E_(2))=P_(1)* P_(2)\\\\P(E_(2))=(1)/(6)* (1)/(5)=(1)/(30)`

c)

The no of ways the comics can perform are all the possible arrangements of 6 comics which are equal to
`6!=720`

Out of these arrangements only one will correspond to the order of D,C,E,A,B,F thus the probability becomes

`P(E_(3))=(1)/(720)`

d)

The probability that comic B or comic F will perform third is given by

`P(E_(4))=P_(1)+P_(2)\\\\P(E_(4))=(1)/(6)+(1)/(6)=(1)/(3)`

where
`P_(1)` is the probability that comic B will perform third and

`P_(2)` is the probability that comic F will perform third