Probability - Coin toss - Conditional Probability

Question

I have been unable to crack this question:

A fair coin is tossed $n$ times. What is the probability of heads on the first toss given that $r$ heads were obtained in the $n$ tosses?

From: http://personal.vu.nl/h.c.tijms/ExamQuestionsUP.pdf

The solution uses conditional probability (which I am unfamiliar with, however, did search to understand). If anyone could explain that would be awesome.

Thanks!

Answer 1

Hint:

It can be rephrased like this:

If there are $n$ balls in an urn and exactly $r$ of them are red, then - if you pick out one ball - what is the probability that the ball is red?

Answer 2

Here is one approach:

Let $A$ be the event that the first toss is heads. Let $B$ be the event that $r$ heads are obtained in the first $n$ tosses. The number of ways to get $r$ heads when flipping a coin $n$ times is $\binom{n}{r}$. The number of ways to get $r$ heads and also have the first toss be heads is $\binom{n-1}{r-1}$. (Reason: Of the remaining $n - 1$ tosses, we need to select $r - 1$ of them to be heads.) Thus, \begin{align} P(A \mid B ) &= \frac{P(A \cap B)}{P(B)} \\ &= \frac{\binom{n-1}{r-1}}{\binom{n}{r}} \\ &= \frac{r}{n}. \end{align}