To solve this problem, we'll use the concept of the binomial probability distribution.
We're given:
- The probability that a human resource manager says job applicants should follow up within two weeks, [tex]\( p = 0.61 \)[/tex].
- The number of human resource managers selected, [tex]\( n = 20 \)[/tex].
- We need to find the probability that exactly 16 of them say that job applicants should follow up within two weeks, hence [tex]\( k = 16 \)[/tex].
The formula for the binomial probability is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
Here, [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, calculated as:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
By substituting the given values:
- [tex]\( n = 20 \)[/tex]
- [tex]\( k = 16 \)[/tex]
- [tex]\( p = 0.61 \)[/tex]
We get the probability:
[tex]\[ P(X = 16) = \binom{20}{16} (0.61)^{16} (1-0.61)^{4} \][/tex]
By calculating each part:
1. Binomial coefficient [tex]\( \binom{20}{16} \)[/tex]:
[tex]\[ \binom{20}{16} = \frac{20!}{16!(20-16)!} = \frac{20!}{16! \cdot 4!} \][/tex]
2. Term involving [tex]\( p \)[/tex]:
[tex]\[ (0.61)^{16} \][/tex]
3. Term involving [tex]\( 1-p \)[/tex]:
[tex]\[ (0.39)^4 \][/tex]
So the final expression for the probability is:
[tex]\[ P(X = 16) = \frac{20!}{16! \cdot 4!} \cdot (0.61)^{16} \cdot (0.39)^4 \][/tex]
Plugging all of this into a calculator or appropriate software, the result is:
[tex]\[ P(X = 16) \approx 0.041193583125192 \][/tex]
Rounding to four decimal places, we get:
[tex]\[ P(X = 16) \approx 0.0412 \][/tex]
Therefore, the probability that exactly 16 out of 20 randomly selected human resource managers say that job applicants should follow up within two weeks is approximately 0.0412.
