Question

An experiment consists of rolling ten fair dice each numbered 1 to 6. The outcome is the sequence of numbers obtained. Answer the following questions:

1. What is the size of the sample space of this experiment?

2. Let random variables X₁, X₂, X₃, X₄, X₅, X₆ be defined as : X_i is the number of i’s obtained among the 10 numbers

   that show up. What is the expectation, E(X₁ + X₂ + X₃ + X₄ + X₅ + X₆)?

3. Let A be the event that exactly five 4’s are obtained, and B be the event that exactly five 3’s are obtained. What

   is P (A | B)? Are A, B independent?

Answer 1

a) The size of the sample space for the experiment is computed here as:
= 6*6*..... 10 times as it is rolled 10 times
= 6¹⁰
= 60466176

Therefore 60466176 is the size of the sample space here.

b) The expected value here is computed as:

because we know that only one of the 6 outcomes can come in any dice throw, therefore the sum of the number of times each one of them occurs is always equal to 10 as there are a total of 10 dice throws here.

c) We are given here that A is the event that exactly five 4’s are obtained, and B is the event that exactly five 3’s are obtained

The probability is computed using Bayes theorem here as:

Therefore 1/3125 is the required probability here.

We first compute P(A) here as:

This is clearly not equal to P(A | B), therefore A and B are not independent as P(A | B) is not equal to P(A)