A) The answer is
38%.
The event has only two possible outcomes: the top mark is earned either by a man or by a woman.
The events are also independent of each other: the top mark earner does not depend on who earned the second highest mark.
Therefore, we are talking about a binomial distribution, in which the probability of success (mark earned by a woman) is
p(W) = 0.38, which means 38%.
B) T<span>he probability that exactly 6 of the 10 top marks were earned by women is
9.34%.</span>
The probability of getting exactly 6 women in 10 marks is given by the formula:
![P(X) = \frac{n!}{k!(n-k)!} p^{k}(1 - p)^{n-k}](https://tex.z-dn.net/?f=P%28X%29%20%3D%20%20%5Cfrac%7Bn%21%7D%7Bk%21%28n-k%29%21%7D%20p%5E%7Bk%7D%281%20-%20p%29%5E%7Bn-k%7D%20%20%20)
where:
n = total number of events = 10
k = number of success we want = 6
p = probability of a succesfull event = 0.38
Substituting the numbers:
![P(W=6) = \frac{10!}{6!(10-6)!} 0.38^{6}(1 - 0.38)^{10 - 6} \\ = \frac{10!}{6!(4)!} 0.38^{6}(0.62)^{4}](https://tex.z-dn.net/?f=P%28W%3D6%29%20%3D%20%5Cfrac%7B10%21%7D%7B6%21%2810-6%29%21%7D%200.38%5E%7B6%7D%281%20-%200.38%29%5E%7B10%20-%206%7D%20%5C%5C%20%0A%3D%20%20%5Cfrac%7B10%21%7D%7B6%21%284%29%21%7D%200.38%5E%7B6%7D%280.62%29%5E%7B4%7D)
P(W = 6) = 210 · 0.00301 · 0.14776
= 0.0934
Hence, the probability of having 6 women earning among the top 10 marks is
0.0934, which means 9.34%
C) We would expect to have
3 women in the top 10.
We would expect that the percentage of the total population is the same of the top 10 marks, therefore:
W = n · p
= 10 · 0.38
= 3.8
Since we cannot have decimals of a physical person, the closest integer is
3.