Find the probability that two friends share the same birth-month.

The probability that two friends share the same birth-month is

(A) $\dfrac{1}{6}$,    (B) $\dfrac{1}{12}$,    (C) $\dfrac{1}{144}$,    (D) $\dfrac{1}{24}$ 

(GATE 1998) 

Answer: (B) $\dfrac{1}{12}$ 

Explanation: 

Sample space, $S = $ 

$\{(Jan, Jan), (Jan, Feb), \ldots, (Jan, Dec), $

$\cdots$    $\cdots$    $\cdots$    $\cdots$    $\cdots$    $\cdots$    $\cdots, $

$(Dec, Jan), (Dec, Feb), \ldots, (Dec, Dec)\}$. 

No. of sample points, $n(S) = 12 \times 12 = 144$. 

Favorable events, $E = \{(Jan, Jan), (Feb, Feb), \ldots, (Dec, Dec)\}$. 

No. of event points, $n(E) = 12$. 

Therefore, the required probability that the two friends share the same birth-month is 

$= \dfrac{n(E)}{n(S)} = \dfrac{12}{144} = \dfrac{1}{12}$.