Question

A professor arrives late to a lecture with probability 1/5. She arrives underprepared with probability 1/2. The probability the professor turns up either late or underprepared or both is 3/5. Are the events that the professor turns up late, and the event that the professor turns up underprepared, independent? Provide both definitions and reasoning.

Answer 1

(a)

To prove that the events:

Event A = Professor turns up late

and

Event B = Professor turns up unprepared

are independent

by definitions :

By Addition Theorem:
P(A + B) = P(A) + P(B) - P(AB)

Given:
P(A) = 1/5

P(B) = 1/2

P(A + B) = 3/5

Substituting, we get:

3/5 = 1/5 + 1/2 - P(AB)

Thus, we get:
P(AB) = 1/10

P(A) X P(B) = 1/5 X 1/2 = 1/10

Thus, we get:
P(AB) = P(A) X P(B)

Thus, Event A and Event B are independent.

Thus, we prove that the Events:Professor turns up late and Professor turns up unprepared are independent by definition.

(a)

To prove that the events:

Event A = Professor turns up late

and

Event B = Professor turns up unprepared

are independent

by reasoning:

The Professor turns up late due to variety of reasons. Professor turns up unprepared due to entirely different reasons. So, both Events are independent.