Question

1. among the contestants in a competition are 45 women and 25 men. if 5 winners are randomly selected,

what is the probability that exactly two are men? (4 pts)

Explanation:

Step1: Calculate the total number of contestants

The total number of contestants is \(45 + 25=70\). The number of ways to choose \(5\) winners out of \(70\) is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 70\) and \(r=5\). So, \(C(70,5)=\frac{70!}{5!(70 - 5)!}=\frac{70\times69\times68\times67\times66}{5\times4\times3\times2\times1}=12103014\)

Step2: Calculate the number of ways to choose exactly 2 men and 3 women

The number of ways to choose \(2\) men out of \(25\) is \(C(25,2)=\frac{25!}{2!(25 - 2)!}=\frac{25\times24}{2\times1} = 300\). The number of ways to choose \(3\) women out of \(45\) is \(C(45,3)=\frac{45!}{3!(45 - 3)!}=\frac{45\times44\times43}{3\times2\times1}=14190\). By the multiplication principle, the number of ways to choose exactly \(2\) men and \(3\) women is \(C(25,2)\times C(45,3)=300\times14190 = 4257000\)

Step3: Calculate the probability

The probability \(P\) that exactly \(2\) men are chosen is \(\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So, \(P=\frac{4257000}{12103014}\approx0.352\)

Answer:

The probability that exactly two are men is approximately \(0.352\)