Question

Let three fair coins be tossed. Let $A =$ {all heads or all tails}, $B =$ {atleast two heads), and $C =$ {atmost two tails). Which of the following events are independent ?

• A. $A$ and $C$
• B. $B$ and $C$
• C. $A$ and $B$
• D. None of these

Answer 1

Answer: (C)

$A$ and $B$

Answer 2

The events can be written explicitly $A = \\{HHH,\, TTT\\}$, $B = \\{HHH,\, HHT,\, HTH,\, THH\\}$ $C = \\{HHH,\, HHT,\, HTH,\, THH,\, HTT,\, THT,\, TTH\\}$ $P(A \cap B) = 1/8$ also $P(A). P(B) = (2/8)(4/8) = 1/8 = P(A \cap B)$ So, $A$ and $B$ are independent. $P(A \cap C) = 1/8$ also $P(A). P(C) = (2/8)(7/8) = 7/32 \ne P(A \cap C)$ So, $A$ and $C$ are dependent. $P(B \cap C) = 4/8$ also $P(B). P(C) = (4/8)(7/8) = 7/16 \ne P(B \cap C)$ So, $B$ and $C$ are dependent.