Question

1. If a student randomly guesses at five multiple-choice questions, find the probability that the student gets exactly three correct. Each question has five possible choices.
2. Say 40% of the class is female. What is the probability that 6 of the first 10 students walking in will be female?

Answer 1

1. Let $X=$ the number of correct answers: $X\sim Bin (n, p).$

Given $n=5, p=\dfrac{1}{5}=0.2, q=1-p=0.8.$

$$P(X=3)=\dbinom{5}{3}(0.2)^3(0.8)^{5-3}=0.0512$$

2. Let $X=$ the number of female: $X\sim Bin (n, p).$

Given $n=10, p=0.4, q=1-p=0.6.$

$$P(X=6)=\dbinom{10}{6}(0.4)^6(0.6)^{10-6}=0.111476736$$