Question

7. A high school is planning to put on the musical West Side Story. There are 20 singers auditioning for the musical. The director is looking for two singers who could sing a good duet.

a. What is the probability that Alicia and Juan are the two singers who are selected by the director? How did you get your answer?
b. The director is also interested in the number of ways to choose a lead singer and a backup singer. What is the probability that Alicia is selected the lead singer and Juan is selected the
backup singer? How did you get your answer?

Answer 1

a) 1/190

b) 1/380

Explanation:

a) Probability of Alicia/Juan getting selected as the first singer = 2/20

Probability of either of them who is left to get selected as the second singer = 1/19

Hence, probability of both Alice and Juan being selected as the singers = (2/20)*(1/19) = 1/190=5.26*10^-3

Another way for this part is:

1/(20C2) = 1/190 (it is a simple case of combination of two from 20)

b) The (a) part included the simple selection but not the arrangement/order in which Alicia and Juan are selected

Considering there are 2 positions (Lead and Backup Singer), there can only be twice as many selection for both (neglecting the order of selection)

So for the probability that only Alicia is selected as the lead singer and Juan is selected the as backup singer we will simply divide the above answer by two

Hence, probability that only Alicia is selected as the lead singer and Juan is selected the as backup singer = (1/190)*(1/2) = 1/380 = 2.63*10^-3

Another way for this part is:

1/(20P2) = 1/380 (it is a simple case of permutation of two from 20)