In a large introductory statistics lecture hall the professo

In a large introductory statistics lecture hall, the professor reports that 56 % of the students enrolled have never taken a calculus course, 30 % have taken only one semester of calculus, and the rest have taken two or more semesters of calculus. The professor randomly assigns students to groups of three to work on a project for the course. You are assigned to be part of a group. What is the probability that of your other two groupmates,

a) neither has studied calculus?

b) both have studied at least one semester of calculus?

c) at least one has had more than one semester of calculus?

Solution

Given never: 56%; one semester: 30%; two or more: the rest

Probability that of your other two groupmates

a) neither has studied calculus

0.56*0.56 = 0.3136

b) both have studied at least one semester of calculus

P(at least one semester) = 1-P(never) = 1 – 0.56 = 0.44; So for both

c) at least one has had more than one semester of calculus

The opposite event of “at least one has more than one semester” is “none of them have more than one semester”, or rather, “both of them have less or equal one semester”. Now: P(one has less or equal one semester) = P(never) + P(one sem.) = 0.56+0.30 = 0.86. So for both of them, the answer is 0.86* 0.86. Finally the answer for the original event that is asked, is 1- 0.86* 0.86 = 0.2604