A fair coin is tossed twice. Let A be the event “a head on the first toss”, B the event “a head on the second toss” and C the event “exactly one head in the two tosses”. Consider the following statements:
(A) A and B are independent.
(B) A and C are independent.
(C) A, B and C are mutually independent.
(D) \(P(C|A\cap B) = 0\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
The sample space is {HH, HT, TH, TT}, and the events are A = {HH, HT}, B = {HH, TH} and C = {HT, TH}, so
\( \displaystyle P(A) = P(B) = P(C) = \frac{1}{2} \)
The common parts are \(A\cap B = \{HH\}\), \(A\cap C = \{HT\}\), \(B\cap C = \{TH\}\), and \(A\cap B\cap C\) is empty:
\( \displaystyle P(A\cap B) = P(A\cap C) = P(B\cap C) = \frac{1}{4},\quad P(A\cap B\cap C) = 0 \)
Three events are mutually independent only when each pair satisfies the product rule and also \(P(A\cap B\cap C) = P(A)P(B)P(C)\).
So each pair has probability \(\frac{1}{4} = \frac{1}{2}\times\frac{1}{2}\), while \(P(A\cap B\cap C) = 0\) against \(P(A)P(B)P(C) = \frac{1}{8}\), and \(P(C|A\cap B) = 0\).
Checking the statements
(A) A and B are independent. — Correct. \(P(A\cap B) = \frac{1}{4} = \frac{1}{2}\times\frac{1}{2}\).
(B) A and C are independent. — Correct. \(P(A\cap C) = P(\{HT\}) = \frac{1}{4} = \frac{1}{2}\times\frac{1}{2}\).
(C) A, B and C are mutually independent. — Incorrect. \(P(A\cap B\cap C) = 0\), but \(P(A)P(B)P(C) = \frac{1}{8}\).
(D) \(P(C|A\cap B) = 0\) — Correct. \(A\cap B\) is HH, which has two heads, not exactly one.