Question

A con man has 3 coins. One coin has been specially made and has a head on each side. A second coin has been specially made, and on each side it has a tail. Finally, a third coin has a head and a tail on it. All coins are of the same denomination. The con man places the 3 coins in his pocket, selects one, and shows you one side. It is heads. He is willing to bet you even money that it is the two-headed coin. His reasoning is that it can’t be the two-tailed coin since a head is showing; therefore, there is a 50-50 chance of it being the two-headed coin. Would you take the bet?

Answer 1

No.

Because probability of double headed (DH) coin = 2/3

Probability of getting head in double headed coin = 1

Probability of getting single head (SH) in coin with head and tail both = 1/2

 

P(selecting two headed coin | head) = P(head in two headed coin/P(head)

                                                                 = P(DH)/{P(DH) + P(SH)}

                                                                 = (1/2 × 1)/{1/2 × 1 + 1/2 × 1/2}

                                                                 = 2/3

                                                                 = 0.67

P(selecting two headed coin | head) = 0.67