Question

In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides.

Answer 1

Let $E_1$ be the event that biased coin (heads on both sides) is selected, $E_2$ be the event that unbiased coin is selected and $A$ be the event that all five tosses show head up.

$\therefore P\left(E_1\right)=\frac{2}{10},P\left(E_2\right)=\frac{8}{10}$

$P\left(A/E_1\right)=1$ and $P\left(A/E_2\right)=\frac{1}{32}$

By using Baye's Theorem, we have

$P\left(E_1/A\right)=\frac{P\left(E_1\right).P\left(A/E_1\right)}{P\left(E_1\right).P\left(A/E_1\right)+P\left(E_2\right).P\left(A/E_2\right)}$

$=\frac{\frac{2}{10}\times 1}{\frac{2}{10}\times 1+\frac{8}{10}\times \frac{1}{32}}=\frac{8}{9}$