ProbabilityHardBloom L3
Question
A certain coin lands head with probability $P$. Let $Q$ denote the probability that when the coin is tossed four times, the number of heads obtained is even. Then
Options
A.There is no value of P , if $Q = \frac{1}{4}$.
B.There is exactly one value of P if $Q = \frac{3}{4}$.
C.There are exactly two values of P , if $Q = \frac{3}{5}$.
D.There are exactly four values of P if $Q = \frac{4}{5}$.
Solution
Sol. [Key Concept: Even number of heads in 4 tosses $= \frac{1}{2}\lbrack 1 + (1 - 2P)^{4}\rbrack$.]
$$\begin{matrix} Q & \ = 1 - \left( \ ^{4}C_{1}P(1 - P)^{3} + \ ^{4}C_{3}P^{3}(1 - P) \right.\ \\ & \ = 1 - \frac{1}{2}\left\lbrack (P + (1 - P))^{4} - \left( ( - P + (1 - P))^{4} \right\rbrack \right.\ \\ & \ = \frac{1}{2}\left( 1 + (1 - 2P)^{4} \right) \end{matrix}$$
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