Question

We have 7 boys and 3 girls in our church choir. There is an upcoming concert in the local town hall. Unfortunately, we can only have 5 youths in this performance. This performance team of 5 has to be picked randomly from the crew of 7 boys and 3 girls. What is the probability that exactly 4 boys are picked in this team of 5?

Answer 1

There is a 41.67% probability that exactly 4 boys are picked in this team of 5.

Explanation:

The order is not important, so we use the combinations formula.

`C_(n,x)` is the number of different combinatios of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

Number of desired outcomes.

Four boys and one girl: So

`C_(7,4)*C_(3,1) = (7!)/(4!(7-4)!)*(3!)/(1!(3-1)!) = 35*3 = 105`

Number of total outcomes:

Combination of five from a set of 10.

So

`C_(10,5) = (10!)/(5!(10-5)!) = 252`

What is the probability that exactly 4 boys are picked in this team of 5?

`P = (105)/(252) = 0.4167`

There is a 41.67% probability that exactly 4 boys are picked in this team of 5.

Answer 2

105/252 = 0.41666...

Explanation:

There are (7C4)(3C1) = (35)(3) = 105 ways to choose exactly 4 boys. There are 10C5 = 252 ways to choose 5 youths, so the probability that a randomly chosen team will consist of exactly 4 boys is ...

105/252

_____

nCk = n!/(k!(n-k!))