Question

There are three coins. One is two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?

Answer 1

Let P( C 1 ), P( C 2 ), P( C 3 ) and P( H ) be the probabilities that are defined below,

The probability that two headed coin is P( C 1 ).

The probability that biased coin is P( C 2 ).

The probability that unbiased on the coin is P( C 3 ).

The probability that head appears on the coin is P( H ).

The probability that the coin is two headed, if it shows head that is P( C 1 H ),

P( C 1 H )= P( C 1 )P( H C 1 ) P( C 1 )P( H C 1 )+P( C 2 )P( H C 2 )+P( C 2 )P( H C 2 ) (1)

The probability for two headed coins is,

P( C 1 )= 1 3

The probability that the head appear on the coin C 1 ,

P( H C 1 )=1

The probability for biased coin is,

P( C 2 )= 1 3

The probability that the head appear on the coin C 2 ,

P( H C 2 )= 3 4

The probability for unbiased coin is,

P( C 3 )= 1 3

The probability that the head appear on the coin C 3 ,

P( H C 3 )= 1 2

Put these values in equation (1),

P( C 1 H )= ( 1 3 ×1 ) ( 1 3 ×1 )+( 1 3 × 3 4 )+( 1 3 × 1 2 ) = 1 3 1 3 + 1 4 + 1 6 = 4 9

Thus, the required probability is 4 9 .