Question

Two coins are in a hat. The coins look alike, but one coin is fair (with probability 1/2 of heads), while the other coin is biased, with probability 1/4 of heads. One of the coins is randomly pulled from the hat, without knowing which of the two it is. Call the chosen coin coin c. (a) coin c is tossed twice, showing heads both times. Given this information, what is the probability that coin c is the fair coin?

Answer 1

Answer:

The coins look alike, but one coin is fair (with probability 1 / 2 of Heads), while the other coin is biased, with probability 1 / 4 of Heads.....................

Answer 2

The probability that coin c is the fair coin is 4/13.

Given,

The number of coins in a hat= 2.

The number of fair coins=1.

Probability of heads in fair coin= $\frac{1}{2}$.

Probability of heads in biased coin= $\frac{1}{4}$.

To find:

The probability that coin c or the chosen coin is the fair coin.

Solution:

Let the probability of picking the fair coin be P(f).

Let the probability of picking the biased coin be P'(f).

P(f)=P'(f)= $\frac{1}{2}$.

Let the probability of getting two heads in a row be P(X).

According to the question, P($\frac{X}{f}$) = $\frac{4}{16}$.

And, P'($\frac{X}{f}$) = $\frac{9}{16}$.

According to Baye's rule,

P($\frac{f}{X}$)= $\frac{P'(\frac{f}{X}) }{P(X)}$

=> P($\frac{X}{f}$) × $\frac{P(f)}{P(X)}$

= 4/13.

Therefore. the answer is 4/13.