Problem
Suppose you have a fair coin and toss it until you get example two heads.
Describe the sample space $\Omega$.
What is the probability that exactly $k$ tosses are required?
Solution
$\Omega = \{HH, HTH, THH, TTHH, THTH, ...\}$
To figure out the probability it takes exactly $k$ tosses, we need to count how many ways there are to get 2 heads in $k$ tosses and divide it by all the ways there are to toss coins in $k$ tosses.
The denominator will be $2^n$.
Consider the numerator. It will be something like:
$1, 2, ..., k-1$ tosses where there is only 1 head followed by the $kth$ toss which will be the final head.
How many ways are there to toss one head in $k-1$ tosses? $k-1 \choose 1$ which is just $k-1$.
So $P(\text{2 heads in exactly k tosses}) = \dfrac {k-1} {2^k}$
Sources
- 36-705 hw1 problem 3