Given:
The bartender start tossing a coin when each customer orders, and if the coin lands on a head, the customer will get a free drink. Assuming the probability of the coin landing heads is p. The first free drink is given to the 5th customer. Assuming no free drink was given between the fifth customer and the 10th. What's the probability of 10th customer get a free drink?
Solution 1:
The probability of the first free drink was given out after 5 tosse= $(1-p)^4 \cdot p$
The probability of the 2nd free drink was given out to 10th customer= $(1-p)^8 \cdot p^2$?
Solution 2:
For every customer the probability of getting free drink is always p?
The probability of any customer getting a free drink is of course $p$. The question is to estimate $p$ given the evidence, and for this, use maximum-likelihood estimation.
Given there is one free drink out of nine, the maximum-likelihood estimate is $p_{MLE} = \frac{1}{9}$.
So the probability the 10th customer will get a free drink is $\frac{1}{9}$.
In the relatively unlikely event that the 10th customer indeed gets a free drink, the new maximum-likelihood estimate would be $p_{MLE} = \frac{1}{5}$.