Question

A student is applying to Harvard and Dartmouth. He estimates that he has a probability of .5 of being accepted at Dartmouth and .3 of being accepted at Harvard. He further estimates the probability that he will be accepted by both is .2. What is the probability that he is accepted by Dartmouth if he is accepted by Harvard? Is the event "accepted at Harvard" independent of the event "accepted at Dartmouth"?

Answer 1

Explanation:

Let P(H) be the probability that he is accepted by Harvard

Let P(D) be the probability that he is accepted by Dartmouth

Let P(HnD) be the probability that he is accepted by Harvard and Dartmouth

Given data

P(H) = 0.3

P(D) = 0.5

P(DnH) = 0.2

To get the probability that he is accepted by Dartmouth if he is accepted by Harvard, can be gotten using the conditional probability formula.

P(D|H) = P(DnH)/P(H)

P(D|H) = 0.2/0.3

P(D|H) = 2/10/÷3/10

P(D|H) = 2/10×10/3

P(D|H) 2/3

b) The two events are independent if the occurrence of an event does not affect the other occurring. For the two events to be independent then;

P(DnH) = P(D)P(H)

Given P(D) = 0.5 and P(H) = 0.3

P(D)P(H) = 0.5 × 0.3

P(D)P(H) = 0.15

And since P(DnH) = 0.2, hence P(DnH) ≠ P(D)P(H)

This means that the event "accepted at Harvard" IS NOT independent of the event "accepted at Dartmouth" since the two values are not equal.