Question

bob and his stepmother wanted to start volunteering together. they found 10 opportunities online, 7 of which involved working with kids. if they randomly chose to apply to 7 of the opportunities, what is the probability that exactly 5 of the chosen opportunities involve working with kids? write your answer as a decimal rounded to four decimal places.

Explanation:

Step1: Calculate total combinations

The total number of ways to choose 7 opportunities out of 10 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 10$ and $r=7$. So $C(10,7)=\frac{10!}{7!(10 - 7)!}=\frac{10!}{7!3!}=\frac{10\times9\times8}{3\times2\times1}=120$.

Step2: Calculate favorable combinations

The number of ways to choose 5 opportunities involving kids out of 7 (in - kid opportunities) and $7 - 5=2$ opportunities not involving kids out of $10 - 7 = 3$ (non - in - kid opportunities).
The number of ways to choose 5 out of 7 in - kid opportunities is $C(7,5)=\frac{7!}{5!(7 - 5)!}=\frac{7!}{5!2!}=\frac{7\times6}{2\times1}=21$.
The number of ways to choose 2 out of 3 non - in - kid opportunities is $C(3,2)=\frac{3!}{2!(3 - 2)!}=\frac{3!}{2!1!}=3$.
The number of favorable combinations is $C(7,5)\times C(3,2)=21\times3 = 63$.

Step3: Calculate probability

The probability $P$ is the number of favorable combinations divided by the total number of combinations. So $P=\frac{63}{120}=0.5250$.

Answer:

0.5250