Probability rolling a die

Question

You roll a die repeatedly until you observe a 6 and then you stop. Let N denote the number of rolls you made before you stop. a) What is the probability of N = 10? b) What is the probability of observing 1 in the first roll, if we know that N = 10?

Answer 1

For each roll, the probability of getting a 6 is P(6)=1/6 and P(not 6)=5/6.

The tosses are independent, so their probabilities are multiplied.

To end at N=10, you had 9 instances of P(not 6) and one of P(6) at the end.

The probability of that sequence is

P(6) times P(not 6)^9 = (1/6) (5/6)^9 = (0.167) (0.833)^9 = 0.032 = 3,2%

We know that the first roll could not be 6, as that would stop the process.

The first roll could be any other number from 1 to 5. So the probability of getting

a 1 on that first roll, given that it cannot be a 6 is 1/5 = 20%.