Step 1 :An experiment consists of tossing 7 fair (not weighted) coins, except one of the 7 coins has a head on both sides. We are asked to compute the probability of obtaining exactly 5 heads.
Step 2 :We can divide the problem into two cases. Case 1: The two-headed coin is among the 5 heads. Case 2: The two-headed coin is not among the 5 heads.
Step 3 :For case 1, we need to choose 4 heads from the remaining 6 coins (which are fair), and the remaining 2 coins will be tails.
Step 4 :For case 2, we need to choose 5 heads from the remaining 6 coins (which are fair), and the remaining 1 coin will be a tail, and the two-headed coin will also be a tail.
Step 5 :The total number of outcomes when tossing 7 coins is \(2^7\), but since one coin is two-headed, the total number of outcomes is \(2^6\).
Step 6 :We can calculate the probability for each case and then add them together to get the final probability.
Step 7 :Let's correct this and calculate the probability again.
Step 8 :The total number of outcomes is \(2^6 = 64\).
Step 9 :For case 1, the number of outcomes is \(\binom{6}{4} = 15\).
Step 10 :For case 2, the number of outcomes is \(\binom{6}{5} = 6\).
Step 11 :The probability is \(\frac{15+6}{64} = 0.328125\).
Step 12 :Final Answer: The probability of obtaining exactly 5 heads when tossing 7 coins, one of which has heads on both sides, is \(\boxed{0.328125}\).