Question

A magician has 20 coins in his pocket. Twelve of these coins are normal fair coins (with one head and one tail) and eight are defective coins with heads on both sides. The magician randomly draws a coin from his pocket and flips it. Given that the flipped coin shows a head, what is the probability that it is defective?

Select one:

4/7

8/20

1

1/2

Answer 1

A magician has 20 coins in his pocket. Twelve of these coins are normal fair coins (with one head and one tail) and eight are defective coins with heads on both sides.

Therefore, P(fair coin) = 12/20 and P(defective coin) = 8/20

P(head | fair coin) = 1/2 and P(head | defective coin) = 2/2 = 1

We want to find, P(defective coin | head)

First we find, P(head)

P(head) = [ P(head | defective coin) * P(defective coin) ] + [ P(head | fair coin) * P(fair coin) ]

=> P(head) = [ 1 * 8/20 ] + [ 1/2 * 12/20 ]

=> P(head) = 0.4 + 0.3

=> P(head) = 0.7

Now, we find,

P(defective coin | head)

= [ P(head | defective coin) * P(defective coin) ] / P(head)

= 0.4 / 0.7

= (4/10) / (7/10)

= 4 / 7

=> P(defective coin | head) = 4 / 7

Answer : 4/7