First slide

Theorems of probability

Question

A bag contains $(2 n + 1)$ coins. It is known that n of these coins have a head on both sides where as the rest of the coins are fair. A coin is picked up at random form the bag and is tossed. If the probability that the results in a head is $\frac{31}{42},$ then the value of n is

Moderate

A

7
B

10
C

12
D

8

Solution

$Given, n coins have head on both sides and (n + 1) coins are fair coins.$
$Let E_{1} = Event that an unfair coin is selected$
$E_{2} = Event that a fair coin is selected$
$\begin{array}{l} H = Event that the toss result in a head \\ P (H) = P (E_{1}) \cdot P (H / E_{1}) + P (E_{2}) \cdot P (H / E_{2}) \\ \frac{31}{42} = \frac{n}{2 n + 1} \cdot 1 + \frac{n + 1}{2 n + 1} \cdot \frac{1}{2} \Rightarrow n = 10 \end{array}$

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