Question

Suppose we keep rolling a tetrahedral die (with faces marked as 1, 2, 3, 4) till an even number appears for the first time.

(a) Give a precise description of the sample space.

(b) Give the probability of each elementary outcome (each element of the sample space).

(c) Find the probability of an even number appearing for the first time at the nth roll.

(d) Find the probability of an even number appearing for the first time no later than the nth roll.

Answer 1

(a)

Die has four faces marked with 1, 2, 3 and 4 respectively. So the possible sample space is

S = {1, 2, 3, 4}

(b)

Since each of the four outcomes are equally likely so the probability of each elementary outcome is

P = 1/4

(c)

Out of 4 outcomes, 2 outcomes are even so probability of getting even number in any roll is

p = 2/4 = 0.50

The probability of not getting even number is

1 - p = 0.50

Since each roll is independent from other so the probability of getting odd numbers in each of the first (n-1) rolls is

The probability of an even number appearing for the first time at the nth roll is

It is pdf of Geometric distribution with n=1, 2, 3, ....

(d)

Let N is a random variable shows the number of trials needed to get first even number. The probability of an even number appearing for the first time no later than the nth roll is

Right hand side is Geometric series with first term a = 0.5 and common ratio r = 0.5. So,