# 4. (5pts) At MCPHS university,

**4.** (5pts) At MCPHS university, there are threemathematics courses: Algebra (A), Calculus (C), and Statistics (S).Records show that 45% of students take A, 35% take C, 30% take S,10% take both A and C, 8% take A and S, 5% take C and S, and 3%take A, C, and S.

a. Find the probability of those who only take A.

b. Find the probability of those who only take one mathematicscourse.

c. Find the probability of those who take at least onemathematics course.

d. Find the probability of those who did not take anymathematics course.

Answer:

a) P(only A) = P(A) – P(A and C) – P(A and S) + P(A, C andS)

= 0.45 – 0.1 – 0.08 + 0.03

= 0.3

b) P(only C) = P(C) – P(C and A) – P(C and S) + P(A, C andS)

= 0.35 – 0.1 – 0.05 + 0.03

= 0.23

P(only S) = P(S) – P(S and A) – P(C and S) + P(A, C and S)

= 0.3 – 0.08 – 0.05 + 0.03

= 0.2

P(only one mathematics course) = P(only A) + P(only C) + P(onlyS)

= 0.3 + 0.23 + 0.2

= 0.73

c) P(at least one) = P(A U C U S)

= P(A) + P(C) + P(S) – P(A and C) – P(A and S) – P(C and S) + P(A,C and S)

= 0.45 + 0.35 + 0.3 – 0.1 – 0.08 – 0.05 + 0.03

= 0.9

d) P(did not take any course) = 1 – P(at least one course)

= 1 – 0.9

= 0.1