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:Probability of any event is defined by[tex]P(E)=\frac{favourable case}{Total cases}[/tex]a)Since any of the six comedians can perform first so the probability that comic D will perform first is given by [tex]\frac{1}{6}[/tex]b)The probability of event that comic C will perform first and comic E will perform fourth is given by[tex]P(E_{2})=P_{1}\times P_{2}\\\\P(E_{2})=\frac{1}{6}\times \frac{1}{5}=\frac{1}{30}[/tex]c)The no of ways the comics can perform are all the possible arrangements of 6 comics which are equal to [tex]6!=720[/tex]Out of these arrangements only one will correspond to the order of D,C,E,A,B,F  thus the probability becomes [tex]P(E_{3})=\frac{1}{720}[/tex]d)The probability that comic B or comic F will perform third is given by[tex]P(E_{4})=P_{1}+P_{2}\\\\P(E_{4})=\frac{1}{6}+\frac{1}{6}=\frac{1}{3}[/tex]where [tex]P_{1}[/tex] is the probability that comic B will perform third and[tex]P_{2}[/tex] is the probability that comic F will perform third