A man has five coins, two of which are double-headed, one is double-tailed, and

Question

A man has five coins, two of which are double-headed, one is double-tailed, and two are normal. He shuts his eyes, picks a coin at random, and tosses it. What is the probability that the lower face of the coin is a head? He opens his eyes and sees that the coin is showing heads: what is the probability that the lower face is a head? He shuts his eyes again, and tosses the coin again. What is the probability that the lower face is a head? He opens his eyes and sees that the coin is showing heads: what is the probability that the lower face is a head? He discards this coin, picks another at random, and tosses it. What is the probability that it shows heads?

Explanation / Answer

1. The two double-headed coins have lower face as head whereas the one double-tailed coin has lower face as tail. The two normal coins have a probability of 1/2 each for the lower face to be head.

Expected number of heads for the lower face = 2 + 0 + 1 = 3

Probabillity = 3/5 = 0.6

2. Since the coin is showing heads, the only way that the lower face is also a head is if the coin is double headed. Note that the coin cannot be double tailed as it showed head. Since there are 2 double headed coins, the probability is

2/4 = 0.5

3. Since the coin cannot be double tailed, it can be either one of the double-headed or the two normal ones.

Probability = (2*1+2*0.5)/4 = 3/4 = 0.75

4. The answer here is the same as 2 i.e 0.5

5. Since the coin has been discarded, there are 2 cases:

(i) The discarded coin was double headed. We are left with one double headed, one double tailed and two normal coins

Probability of showing heads = (1 + 0 + 1) / 4 = 0.5

(ii) The discarded coin was normal. We are left with two double headed, one double tailed and one normal coin

Probability of showing heads = (2 + 0 + 0.5) / 4 = 2.5/4 = 0.625

=> Probability that it shows heads = (0.5+0.625)/2 = 0.5625

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