Sure! Let's solve the problem step-by-step:
1. Understanding the Problem:
- Kendall has 27 job applications in total.
- Among them, 3 are for personal assistant jobs.
- Kendall wants to pick 24 applications to submit today.
- We need to find the probability that exactly 2 out of the 24 picked applications are personal assistant jobs.
2. Setting Up the Problem:
- We need to use the concept of combinations to determine the number of ways to pick jobs.
- We later use these combinations to compute the probability.
3. Step-by-Step Solution:
a. Possible Combinations for Selecting Jobs:
- First, calculate the number of ways to choose 2 personal assistant jobs out of the 3:
[tex]\[
\binom{3}{2} = 3
\][/tex]
b. Selecting the Remaining Jobs:
- Next, we need to select the remaining 22 jobs out of the 24 non-personal assistant jobs. There are [tex]\(24 - 3 = 21\)[/tex] non-personal assistant jobs available:
[tex]\[
\binom{24}{22} = 276
\][/tex]
c. Total Combinations:
- Finally, calculate the total number of ways to pick any 24 jobs out of the 27:
[tex]\[
\binom{27}{24} = 2925
\][/tex]
4. Calculating the Probability:
- The probability can be found by dividing the favorable combinations by the total combinations:
[tex]\[
\text{Probability} = \frac{\binom{3}{2} \times \binom{24}{22}}{\binom{27}{24}} = \frac{3 \times 276}{2925}
\][/tex]
- Simplifying the fraction:
[tex]\[
\frac{3 \times 276}{2925} = 0.2831
\][/tex]
Hence, the probability that exactly 2 of the 24 picked applications are for personal assistant jobs is 0.2831 (rounded to four decimal places).
