Discussion
A coin is biased so that head is twice as likely to occur as a tail. If a coin is tossed four times, what is the probability of getting two tails and two heads ?
Answer: B
The probability is 8/27.
Since a head is twice as likely as a tail, \(P(H) = 2/3\) and \(P(T) = 1/3\). Two heads and two tails in four tosses is a binomial event:
\[ \begin{aligned} P &= \binom{4}{2}\left(\tfrac{2}{3}\right)^{2}\left(\tfrac{1}{3}\right)^{2} \\ &= 6 \cdot \tfrac{4}{81} = \tfrac{24}{81} = \tfrac{8}{27} \end{aligned} \]
3/8 is the answer for a fair coin (\(6/16\)). 4/27 (that is, 12/81) counts only three of the six orders of HHTT, and 1/8 has no basis here.
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