Consider an experiment of tossing 3 coins simultaneously. Define the following events:
E = [Three heads or three tails appear]
F = [At least two heads appear] and
G = [At most two heads appear]
Choose the correct option:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → E and F are independent events
Two events are independent when \(P(X \cap Y) = P(X)\,P(Y)\). The sample space has 8 equally likely outcomes.
\(E = \{HHH, TTT\}\)
\(F = \{HHH, HHT, HTH, THH\}\)
\(G = \text{every outcome except } HHH\)
\( \displaystyle P(E) = \frac{2}{8} = \frac{1}{4},\quad P(F) = \frac{4}{8} = \frac{1}{2},\quad P(G) = \frac{7}{8} \)
E and F: \(E \cap F = \{HHH\}\)
\( \displaystyle P(E \cap F) = \frac{1}{8} = \frac{1}{4} \times \frac{1}{2} = P(E)\,P(F) \)
So E and F are independent, and they are not dependent.
E and G: \(E \cap G = \{TTT\}\)
\( \displaystyle P(E \cap G) = \frac{1}{8},\quad P(E)\,P(G) = \frac{1}{4} \times \frac{7}{8} = \frac{7}{32} \)
These differ, so E and G are not independent.
F and G: \(F \cap G = \{HHT, HTH, THH\}\)
\( \displaystyle P(F \cap G) = \frac{3}{8},\quad P(F)\,P(G) = \frac{1}{2} \times \frac{7}{8} = \frac{7}{16} \)
These differ, so F and G are not independent.
NCERT (2026-27 edition):
Independent Events
