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  4. A biased coin has probability p of showing up heads. If it is tossed five times, the probability of exactly two heads is the same as the probability of exactly one head. The probability of exactly three heads in five tosses, is

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Q. A biased coin has probability p of showing up heads. If it is tossed five times, the probability of exactly two heads is the same as the probability of exactly one head. The probability of exactly three heads in five tosses, is

Probability - Part 2

Solution:

${ }^5 C _2 P ^2(1- P )^3={ }^5 C _1 P (1- P )^4 $
[using $P ( X - r )={ }^n C _{ r } p ^{ r } q ^{ n - r }$ ]
$2 P =1- P \Rightarrow P =\frac{1}{3} $
$P ( E )={ }^5 C _3\left(\frac{1}{3}\right)^3 \cdot\left(\frac{2}{3}\right)^2=\frac{40}{243}$