**Binomial distribution**

The binomial distribution is the distribution of the results of a binomial experiment. If the variable follows a binomial distribution, it can be denoted

Here *n* is the number of trials and *p* is the probability of an event occurring in one trial.

The Binomial distribution expected value E and standard deviation D

**Example 1**

We roll a dice 3 times.

Find the distribution of the variable and the expected value for the event "Number of sixes with three rolls"

Solution

This is a binomial experiment in which the number of trials is n = 3 and the probability of a single event is p = 1/6.

There may be 0, 1, 2 or 3 sixes with 3 rolls. Probabilities

Expected value

**Example 2**

It is estimated that 1% of the total population meets the criteria for psychopathy.

a) What is the probability that an aircraft with 420 passengers has exactly 5 psychopaths?

b) find the expected value of the variable.

This is a binomial experiment with 5 trials. The probability of a single event is 0,01

a) Probability

b) Expected value

**Example 3**

Let it be

Illustrate the distribution of a random variable with a bar graph.

First, we calculate the probabilities using the binomial experiment formula and form a distribution.

Since the probability is less than 0,5, the distribution is weighted to the left.

**Example 4**

It is assumed that 60 children will be born in a municipality next year. The sexes of the children are independent of each other, and the probability of having a boy is 0,513. What distribution do the numbers of girls and boys follow? What is the expected value of the number of boys and what is the number of girls? What is the probability of having exactly the same number of girls and boys?

(YO1999K Long Mathematics)

Solution

Birth can be considered a binomial experiment, i.e., numbers follow a binomial distribution. Let X be the number of girls and Y the number of boys.

Expected values

In order to have exactly the same number of girls and boys, both must be born with 30 each. It is enough to calculate one probability. We calculate the probability that 30 girls will be born.

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