tmua.me Tau Paper 1 · Question 20

Seven coins are tossed independently. Their seven probabilities of heads are all distinct and lie strictly between $0$ and $1$ .

For each $k\in\{0,1,\ldots,7\}$ , the probability of exactly $k$ heads equals the probability of exactly $7-k$ heads. One coin has probability of heads $\frac13$ , and the probability that all seven show heads is $\frac1{576}$ .

Which of the following can be the probability of exactly one head?

\dfrac{41}{1296}

\dfrac{13}{384}

\dfrac1{40}

\dfrac1{32}

\dfrac{31}{1152}

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tmua.me Tau Paper 1 · Q20