A biased coin in which the head is 3 times as likely to occur as tail, is tossed twice. If the probability distribution of the number of tails is given by:
| X | 0 | 1 | 2 |
| P(X = x) | \(\displaystyle\frac{9}{16}\) | m | n |
The value of \(\displaystyle\frac{m}{n}\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 6
Since a head is 3 times as likely as a tail and P(H) + P(T) = 1:
\( \displaystyle P(H) = \frac{3}{4},\quad P(T) = \frac{1}{4} \)
X = 1 means exactly one tail (HT or TH); X = 2 means both tails (TT). The two tosses are independent:
\( \displaystyle m = P(X = 1) = 2\times\frac{3}{4}\times\frac{1}{4} = \frac{6}{16} \)
\( \displaystyle n = P(X = 2) = \frac{1}{4}\times\frac{1}{4} = \frac{1}{16} \)
\( \displaystyle \frac{m}{n} = \frac{6/16}{1/16} = 6 \)
Check: \(\frac{9}{16} + \frac{6}{16} + \frac{1}{16} = 1\).