Question

20 students signed up for the talent show. 5 students plan to sing, 6 students plan to dance, and the remaining students plan to play an instrument. The director of the show will randomly choose the order of the first two acts. Calculate the probabilities of each pair of dependent events in parts a to e.

Answer 1

To calculate the probabilities of each pair of dependent events, we need to consider the different possibilities based on the information provided:

Total number of students = 20

Number of students singing = 5

Number of students dancing = 6

Number of students playing an instrument = 20 - (5 + 6) = 9

Now, let's calculate the probabilities for each pair of dependent events:

a) Probability that the first student chosen sings (P(Sing)):

P(Sing) = Number of students singing / Total number of students

P(Sing) = 5 / 20

P(Sing) = 1/4

b) Probability that the second student chosen also sings (P(Sing|Sing)):

Since the first student has already been chosen and has sung, there are now 4 students left who sing out of the remaining 19 students:

P(Sing|Sing) = Number of remaining students who sing / Remaining total number of students

P(Sing|Sing) = 4 / 19

c) Probability that the first student chosen sings (P(Sing)) and the second student chosen dances (P(Dance)):

P(Sing) = 5 / 20 (as calculated in part a)

P(Dance) = 6 / 19 (since there are 6 students who dance out of the remaining 19 students)

To find the joint probability:

P(Sing and Dance) = P(Sing) * P(Dance)

P(Sing and Dance) = (5/20) * (6/19)

d) Probability that the first student chosen sings (P(Sing)) and the second student chosen plays an instrument (P(Instrument)):

P(Sing) = 5 / 20 (as calculated in part a)

P(Instrument) = 9 / 19 (since there are 9 students who play an instrument out of the remaining 19 students)

To find the joint probability:

P(Sing and Instrument) = P(Sing) * P(Instrument)

P(Sing and Instrument) = (5/20) * (9/19)

e) Probability that the first student chosen dances (P(Dance)) and the second student chosen plays an instrument (P(Instrument)):

P(Dance) = 6 / 20 (as there are 6 students who dance)

P(Instrument) = 9 / 19 (as calculated in part d)

To find the joint probability:

P(Dance and Instrument) = P(Dance) * P(Instrument)

P(Dance and Instrument) = (6/20) * (9/19)

These calculations give you the probabilities for each pair of dependent events based on the given information about the students' talents.