Coin toss and geometrical distribution

Question

Can we prove that random variable $X$ that counts the number of coin tosses until the first head/tail appear is geometrically distributed? I cannot seem to find such a proof anywhere. Is it even proven?

Answer 1

That's precisely the definition of a Geometric distribution. A random variable $X$ is said to be Geometrically distributed if it describes an event of success on some $k$-th trial. Therefore, if the probability of success is $p$,

$$P(X=k) = p\times(1-p)^{k-1}$$

that is, you lose the first $k-1$ rounds and win on the $k$-th round.

In the case of a coin toss, the probability of success is $p= \frac{1}{2}$. And yes, the event you have described in the question of counting the coin tosses until you see your first head (your definition of success) is indeed Geometric distribution.