# Heads and tails Probability question

I have two coins in my pocket. One of them is normal. One side heads one side tails. The others both sides are tails. So if I pick a random coin from my pocket and the side I see is "tails" what is the probability that the other side is also "tails".

Does the the probability change when I pick one coin?

I'm sorry if this is asked before but I'm not sure about the theory name or how to search for.

• What do you mean by "Does the probability change when I pick one coin?" ? – augurar Aug 9 '15 at 2:24
• My reasoning was that once you pick a coin and see it. The other side can be either heads or tails. So after picking it up I thought it was 50/50 – Sinan Aug 9 '15 at 8:05
• @Sinan this is a very common quirk of our human minds. It's a good example of how we're not really geared to reason about probability at all. I recommend you look at the Monty hall problem e.g. youtube.com/watch?v=4Lb-6rxZxx0. It confused some of the greatest mathematicians, including Erdos! – Colm Bhandal Aug 9 '15 at 15:21

EDIT: New question is a question about conditional probability. The trick is to apply Bayes' theorem. That is $P(A \mid B) = P(A)P(B\mid A)/P(B)$. So say event $A$ is picking the two tails coin. And say event $B$ is seeing a tails. Then $P(A) = \frac12$ and $P(B) = \frac34$ and $P(B\mid A) = 1$, because we're guaranteed to see tails on the two tails coin. So $$P(A \mid B) = P(A)P(B\mid A)/P(B) = P(A)/P(B) = \frac12 / \frac34 = \frac23$$ This is the probability of your coin being the one with two tails, given that you see tails.
Intuitively this result makes sense. $2$ out of $3$ of the tails are on the two tails coin. The other tails only makes up $1$ out of the total possible $3$.
There are three possible "tails" you could be looking at. Two of these have "tails" on the reverse. The other one has "heads" on the reverse. Each of these are equally likely. So the probability that the other side is "tails" is $2/3$.