# Flipping either a biased or unbiased coin

Here's a question I was faced with recently:

I have an unbiased coin with one head side and one tail side, and a biased coin with two tail sides. I choose an coin whilst blindfolded, and flip the coin. The result is a tail. What is the chance that the coin is biased?

What is obvious is that I have a 50% chance of picking the biased coin to begin with, but what I am not sure about is whether or not the result of the flip affects the chance that I picked the biased coin.

Any help?

• Let $T$ be the event we get a tail, and let $B$ be the event the coin is biased. We want the conditional probability $\Pr(B|T)$. By the definition of conditional probability, we have $\Pr(B|Y)=\frac{\Pr(B\cap T)}{\Pr(T)}$. Now can you calculate the two probabilities on the right? – André Nicolas Jul 21 '15 at 5:28

This is easier to see if you extend the experiment and pick a coin and flip it say $100$ times. You would expect output like:

$$HT_uTTTT_uHTT_uHT\dots THHT$$

with a probability table:

$$\begin{array} {c|c} H&0.25\\ T_u&0.25\\ T&0.5 \end{array}$$

If you throw a tails, $2/3$ times it will come from the biased coin, so if you say 'biased' every time tails appear, you will be right $2/3$ times.

Hint:

You'd better write down all the events, e.g. $B=\{ \text{the coin is biased}\}$ and $R= \{\text{the result is a tail}\}$.

Thus, you need to find the probability $$\Pr(B\mid R)=\dfrac{\Pr( B\cap R)}{\Pr(R)}.$$

Just a hint to find $\Pr( R)$. How can we get tails? One way is to choose the biased coin and then get tails and the other way is to choose the unbiased coin and then get tails again. Try to translate this information in the language of mathematics.

• This isn't for an assignment or anything :) mind helping me out just a bit more? XD – Lucas - Better Coding Academy Jul 21 '15 at 5:36
• I tried to give you a hint. – thanasissdr Jul 21 '15 at 5:43