**Normal distribution**

The normal distribution, or Gaussian distribution, or Gaussian curve, is a continuous probability distribution.

The normal distribution is symmetric on both sides of the expectation value. If a random variable follows a normal distribution, then we denote

Where 𝞵 is the expected value and 𝞼 is the standard deviation.

**Example 1**

A hair dryer operates without damage for a period of time, which is normally distributed with an expected value of 15,2 months and a standard deviation of 2,5 months. The dryer has a one-year warranty.

a) What percentage of dryers are covered by warranty repairs?

b) What percentage of dryers operate without damage for more than 18 months?

(YOS2015 Short Mathematics)

Solution

Time for which the dryer operates = *t*. The variable follows the normal distribution N (15.2; 2.5). The unit is a month.

Let's normalise. That is, the values are compared to a normal distribution with an expected value of 0 and a standard deviation of 1.

a) The warranty period is 12 months. Let’s see how many standard deviations this is from the expected value. (That is, normalise)

In the normalised normal distribution, we want to find out the accumulated probability up to -1,28.

Since the distribution is symmetric on both sides of the expected value, the corresponding positive value can be retrieved from the table and subtracted from 1.

About 10% of the dryers will be under warranty repair.

b) Normalise the value 18, that is, find out how many standard deviations it is from expected value.

An area after 1,12 is now what we want. Retrieve a value from the table up to 1,12 and subtract it from 1.

13% of dryers last more than 18 months.

### Approximation of the binomial distribution by the normal distribution

If the number of trials in a binomial experiment is large, it can be approximated by a normal distribution.

**Example 2**

A dice is rolled 1200 times. What is the probability of getting 180 to 220 sixes?

Find the expected value and the standard deviation and perform an approximation.

Using a normal distribution, determine the probability between 180 and 220.

In the binomial distribution, the values are integers and in the normal distribution, they are real numbers, a so-called continuity correction is made.

We calculate the normal distribution with Geogebra. A probability of about 94,4% has been accumulated at 220,5 and about 5,6% at 179,5

This results in a probability, coming from 180 to 220 sixes per 1200 rolls, of about 88.8%

In the first row, the binomial probability is calculated directly by Geogebra. This value is also 88.8%.

Geogebra (as well as other calculators / applications) can easily calculate binomial probabilities with a large number of trials.

**Turn on the subtitles if needed**

**Exercises**

1. The mean of one entrance exam was 60 and the standard deviation was 12. The results followed the normal distribution.

a) What was the score required for selection if 50% of the candidates were selected.

b) What was the score required for selection if 30% of the candidates were selected.

c) How many scored more than 65 points in the tests

2. Simo-Elmeri was late for school every day. His time of delay followed the normal distribution. The mean was five minutes and the standard deviation was 2 minutes.

a) What is the probability that Simo-Elmeri was more than seven minutes late?

b) What is the probability that Simo-Elmeri was less than four minutes late?

c) What is the probability that Simo-Elmeri was more than four but less than seven minutes late?

3. A dice is rolled 810 times. What is the probability of rolling 410 to 420 even numbers?

4. It is known that 85% of the 385 guests invited to a party will arrive. The party has 333 seats. What is the probability that there are sufficient enough seats?