Question

5 coins are put in a bag. 2 of the coins are weighted with the probability of flipping heads being three times as great than the probability of flipping tails; the remaining coins are fair. One of these coins is selected at random and then flipped once. What is the probability that a weighted coin was selected given that heads was flipped?

Answer 1

total number of coins , n = 5

unfair coins = 2

fair coins = 3

P[ Heads | unfair coin ] = 3*P[ Tails | unfair coin ]

Also, P[ Heads | unfair coin ] + P[ Tails | unfair coin ] = 1

3*P[ Tails | unfair ] + P[ Tails | unfair ] = 1

4*P[ Tails | unfair ] = 1

P[ Tails | unfair ] = 1/4

P[ Heads | unfair coin ] = 3*(1/4)

P[ Heads | unfair coin ] = 3/4

P[ Heads | fair coin ] = 1/2

P[ unfair coin ] = 2/5

P[ fair coin ] = 3/5

P[ Heads ] = P[ Heads | unfair coin ]*P[ unfair coin ] + P[ Heads | fair coin ]*P[ fair coin ]

P[ Heads ] = (3/4)*(2/5) + (1/2)*(3/5)

P[ Heads ] = 3/10 + 3/10

P[ Heads ] = 3/5

What is the probability that a weighted coin was selected given that heads was flipped?

P[ unfair coin | Heads ] = P[ Heads | unfair coin ]*P[ unfair coin ]/ P[ Heads ]

P[ unfair coin | Heads ] = (3/4)*(2/5) / (3/5)

P[ unfair coin | Heads ] = (3/10) / (3/5)

P[ unfair coin | Heads ] = 1/2

P[ unfair coin | Heads ] = 0.5