All the fourth-graders in a certain elementary school took a standardized test. A total of 85% of the students were found to be proficient in reading, 78% were found to be proficient in mathematics, and 65% were found to be proficient in both reading and mathematics. A student is chosen at random.

Question

a. What is the probability that the student is proficient in mathematics but not in reading?
b. What is the probability that the student is proficient in reading but not in mathematics?
c. What is the probability that the student is proficient in neither reading nor mathematics?

Answer 1

(a) Let R be the event that a student is proficient in reading, and let M be the event that a student is proficient in mathematics. We need to find . Now P(M) = P(R ? M) + (this can be seen from a Venn diagram). We know that P(M) = 0.78 and P(R ? M) = 0.65. Therefore = 0.13.



(b) We need to find . Now P(R) = P(R ? M) + (this can be seen from a Venn diagram).

We know that P(R) = 0.85 and P(R ? M) = 0.65. Therefore = 0.20.



(c) First we compute P(R M):

P(R M) = P(R) + P(M) ? P(R ? M) = 0.85 + 0.78 ? 0.65 = 0.98.

Now = 1?P(R M) = 1?0.98 = 0.02.