Question

franklin is filling out summer job applications, so that he can save money to buy a mountain bike. of the 23 jobs hes applying for, 7 are personal assistant jobs and 14 are lifeguarding jobs. if franklin randomly picks 9 applications to submit today, what is the probability that 2 of the chosen applications are for personal assistant jobs and 7 are for lifeguarding jobs? write your answer as a decimal rounded to four decimal places.

Explanation:

Step1: Calculate total ways to pick 9 applications

We use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 23$ and $r=9$. So the total number of ways to pick 9 applications out of 23 is $C(23,9)=\frac{23!}{9!(23 - 9)!}=\frac{23!}{9!14!}=817190$.

Step2: Calculate ways to pick 2 personal - assistant jobs

There are 7 personal - assistant jobs and we want to pick 2. Using the combination formula, $C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7!}{2!5!}=21$.

Step3: Calculate ways to pick 7 lifeguarding jobs

There are 14 lifeguarding jobs and we want to pick 7. Using the combination formula, $C(14,7)=\frac{14!}{7!(14 - 7)!}=\frac{14!}{7!7!}=3432$.

Step4: Calculate favorable ways

The number of favorable ways is the product of the number of ways to pick 2 personal - assistant jobs and 7 lifeguarding jobs. So the number of favorable ways is $C(7,2)\times C(14,7)=21\times3432 = 72072$.

Step5: Calculate probability

The probability $P$ is the number of favorable ways divided by the total number of ways. So $P=\frac{72072}{817190}\approx0.0882$.

Answer:

0.0882