A bag contains (2 n+1) coins. It is known that n of these coins have a head on both sides, whereas the remaining ( n +1) coins are fair. A coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is 31 / 42, then n is equal to

Step-by-step solution

Total number of coins =2 n +1 Consider the following events: E ₁= Getting a coin having head on both sides from the bag. E ₂= Getting a fair coin from the bag A = Toss results in a head Given: P ( A )= 31 42 , P ( E ₁ )= n 2 n +1 and P ( E ₂ )= n +1 2 n +1 Then, P ( A )= P ( E ₁ ) P ( A / E ₁ )+ P ( E ₂ ) P ( A / E ₂ ) 31 42 = n 2 n+1 1+ n+1 2 n+1 1 2 31 42 = n 2 n+1 + n+1 2(2 n+1) 31 42 = 3 n+1 2(2 n+1) 31 21 = 3 n+1 2 n+1 n=10