Question

I roll a fair die repeatedly until a number larger than 4 is observed. If N is the = 1, 2, 3, .... total number of times that I roll the die, find P(N = k) where k How many trials we will need on average?

Answer 1

To find P(N = k), we need to calculate the probability that the first 4 rolls are not larger than 4, and the kth roll is larger than 4.

The probability that any given roll is larger than 4 is 2/6 = 1/3. Therefore, the probability that the first k-1 rolls are not larger than 4 and the kth roll is larger than 4 is
`(2/3)^((k-1)) * (1/3)`.

So,
`P(N = k) = (2/3)^((k-1)) * (1/3)`.

To find how many trials we will need on average, we can use the formula for the expected value of a geometric distribution: E(N) = 1/p, where p is the probability of success (in this case, rolling a number larger than 4).

So, p = 1/3, and E(N) = 1/p = 3. Therefore, on average, we will need 3 trials to observe a number larger than 4.

To know more about probability refer here