Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A box contains $n$ number of coins, $m$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $1/2$, while it is $2/3$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows a head and the second time it shows tail. What is the probability that the coin drawn is fair.

share|cite|improve this question

What have you done so far? Hint: Without the flips, what is the chance you draw a fair coin? If you draw a fair coin, what is the chance you get the flips specified? If you draw a biased coin, what is the chance?

share|cite|improve this answer


  • $A$ = Even that coin selected is fair

  • $B$ = Coin selected is biased

  • $C$= Even that first results in a head, second results in a tail.

Now use Bayes Theorem to get the answer. Answer is $\frac{9m}{8n+m}$

You have:

  • $P(A)=\frac{m}{n}$ and $P(B)=\frac{n-m}{n}$

  • $P(C/A) = \frac{1}{4}$ and $P(C/B) = \frac{2}{3} \times \frac{1}{3} = \frac{1}{9}$.

  • By Bayes theorem $$P(A/C) = \frac{P(A) \times P(C/A)}{P(A) \times P(C/A) + P(B) \times P(C/B)}=\frac{9m}{8n+m}$$

share|cite|improve this answer
it's Bayes (or Bayes') theorem, from Thomas Bayes; not Baye's – leonbloy Jan 19 '11 at 20:12

I know this isn't really answering your question, but a coin toss, is rarely ever fair. And I'm not talking a few thousandths of a percent, but more like 52% vs 48%. Below makes for fascinating reading:

And remember, always be the tosser!

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.