Question

Three events, A, B, and C, are said to be mutually independent if P(An B) = B P(A) * P(B), P(B n C) = P(B) x P(C), P(ANC) = P(A) * P(C), P(An B nC) = P(A) P(B) x PC) Suppose that a balanced coin is independently tossed two times. Define the following events:

A. Head appears on the first toss.
B. Head appears on the second toss.
C. Both tosses yield the same outcome.
Are A, B, and C mutually independent?

Answer 1

No, events A, B, and C are not mutually independent. The conditions for mutual independence are not satisfied for all possible pairs of events.

Step-by-step explanation:

The events A, B, and C are not mutually independent. To determine if events A, B, and C are mutually independent, we need to check if the probability of the intersection of these events is equal to the product of the probabilities of the individual events.

For event A: P(A) = 0.5 because there is a 0.5 probability of getting a head on the first toss of a balanced coin.

For event B: P(B) = 0.5 because there is a 0.5 probability of getting a head on the second toss of a balanced coin.

For event C: P(C) = 0.5 because there is a 0.5 probability of both tosses yielding the same outcome (both heads or both tails).

Now, let's check if the conditions for mutual independence are satisfied for all possible pairs of events:

P(A ∩ B) = P(A)P(B) = (0.5)(0.5) = 0.25

P(B ∩ C) = P(B)P(C) = (0.5)(0.5) = 0.25

P(A ∩ C) = P(A)P(C) = (0.5)(0.5) = 0.25

P(A ∩ B ∩ C) = P(A)P(B)P(C) = (0.5)(0.5)(0.5) = 0.125

Since P(A ∩ B ∩ C) is not equal to P(A ∩ B)P(C), A, B, and C are not mutually independent.