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 fifthb. comic f will perform first and comic e will perform thirdc. the comedians will perform in the following order: d, f, e, a, b, c.d. comic b or comic c will perform seconda. $square$(type a fraction. simplify your answer.)b. $square$(type a fraction. simplify your answer.)c. $square$(type a fraction. simplify your answer.)d. $square$(type a fraction. simplify your answer.)

Explanation:

Step1: Total permutations of 6 comics

Total possible orders = $6! = 6\times5\times4\times3\times2\times1 = 720$

Step2: Solve part a: Comic D 5th

Fix D in 5th spot. Arrange remaining 5 comics: $5! = 120$
Probability = $\frac{\text{Favorable permutations}}{\text{Total permutations}} = \frac{120}{720} = \frac{1}{6}$

Step3: Solve part b: F 1st, E 3rd

Fix F in 1st, E in 3rd. Arrange remaining 4 comics: $4! = 24$
Probability = $\frac{24}{720} = \frac{1}{30}$

Step4: Solve part c: Exact order D,F,E,A,B,C

Only 1 favorable permutation.
Probability = $\frac{1}{720}$

Step5: Solve part d: B or C 2nd

Favorable cases: B in 2nd ($5! = 120$) + C in 2nd ($5! = 120$) = 240
Probability = $\frac{240}{720} = \frac{1}{3}$

Answer:

a. $\frac{1}{6}$
b. $\frac{1}{30}$
c. $\frac{1}{720}$
d. $\frac{1}{3}$