Question

A fair die is rolled 10 times. What is the average number of even number outcomes?

Answer 1

Concepts and reason

Binomial distribution:

In a series of n independent trials, for each trial if the probability of success is a constant p and the probability of failure is $q=1-p $; then, the probability of x success and obviously failures is given by the Binomial distribution.

Requirements of a binomial experiment are as follows:

• For the given experiment, each trial has two possibilities. That is, it has a success and a failure, which are mutually exclusive outcomes.

• The number of trials in the given experiment is known in advance, and they are fixed.

• The outcomes of the trials are independent.

• The probability of success is unaltered for each trail throughout the experiment.

Fundamentals

Binomial probability of success:

Where n is the number of trials and p is the probability of success for each trial.

Binomial probability of failure:

$p=1-9 $

Where,

.

Binomial distribution probability mass function:

The binomial distribution is the discrete probability distribution and its probability is,

Also,

Where n is the number of trials and p is the probability of success for each trial.

• The mean formula for the binomial distribution is, .

• The standard deviation formula for the binomial distribution is, .

The objective of the problem is determined below:

From the information given, a fair die is rolled 10 times. That is, $n=10 $, and the probability of the outcome is an even number is 0.5. That is, $p=0.5 $, by using the probability and sample size the average number of even number outcomes is obtained.

The average number of even number outcomes is obtained as shown below:

The expected value is,

Ans:

The average number of even number outcomes is 5.