Question

One fair coin and two unfair coins where heads is 5 times as likely as tails are put into a bag. One coin is drawn at random and then flipped twice. If at least one of the flips was tails, what is the probability an unfair coin was flipped?

Answer 1

We are given 3 coins here, one fair coin and two unfair coins.
Therefore P( fair coin) = 1/3 and P(unfair coin) = 2/3

Also, we are know here that:
P( heads | fair coin) = 1/2 = 0.5
P( heads | unfair coin ) = x
P( tails | unfair coin ) = x/5 = 0.2x

Therefore x + 0.2x = 1
x = 5/6

Therefore P( heads | unfair coin) = 5/6

Now for 2 tosses we get here:
P( at least one tail | fair coin) = 1 - P(both heads | fair coin) = 1 - 0.5² = 0.75
P( at least one tail | unfair coin) = 1 - P( both heads | unfair coin) = 1 - (5/6)² = 0.3056

Using law of total probability, we get here:
P( at least one tail) = P( at least one tail | fair coin)P(fair coin) + P( at least one tail | unfair coin)P(unfair coin)

P( at least one tail) = 0.75/3 + 0.3056*(2/3) = 0.4537

Using Bayes theorem now, we get here:

P( unfair coin | at least one tail) = P( at least one tail | unfair coin)P(unfair coin) / P( at least one tail)

= 0.3056*(2/3) / 0.4537

= 0.4490

Therefore 0.4490 is the required probability here.