Binomial experiment

Introduction example

Consider a multiple-choice test of five questions, each with five options. What is the probability you will get three question right by guessing?

For each question, P (“guess right”) = 0,20 and P (“guess wrong”) = 0,80. We make a table of the situation listing all possible options. In the table, V = incorrect and O = correct.

It is noted that the probability of each answer line is a product

The number of favourable response lines is seen in the table and is ten. This can also be calculated without tabulation, i.e. We solve how many ways to choose a 3-element subset from a 5-element set.

Now the probability of getting exactly three question correctly is

Answer: The probability of getting exactly three correctly is 0,051

If the probability of event A does not depend on the results of previous experiments, it is a binomial experiment.

Assume that the probability of event A is p. The experiment is repeated n times. Here

Example 1

On average, 3,5% of the products manufactured by one company have been found to be defective.

a) What is the probability that exactly three of 20 products randomly selected are defective?

b) What is the probability that at least two of 20 randomly selected products are defective?

c) What is the probability that the first of the 20 randomly selected products will be the only defective one?

a) Binomial probability

Answer: Out of 20 products, exactly three are defective with a probability of 0,027

b) It makes sense to calculate the probability using a complementary event.

P (“product is defective”) = 0,035

P (“product is not defective”) = 1 - 0,035 = 0,965

Answer: The probability that at least 2 of the 20 products are defective is about 0,15.

c) This is not a binomial experiment because the defective product is in place. The problem is calculated using a multiplication rule

Example 2

A chessboard has an 8 x 8 grid and is surrounded by a 5 cm wide, grey edge. Every other square on the grid is white and every other black. The size of the board with its edges is 50 cm X 50 cm. 30 rice grains are dropped on the board. What is the probability that the centre of at least 15 rice grains will hit the white square? (YO Spring 2019)

The total area of the chessboard is 50 cm · 50 cm = 2500 cm²

The area of the grid is 40 cm · 40 cm = 1600 cm²

The area of the white squares is half the area of the grid, 800 cm²

The probability that at least 15 rice grains fall into a white box can be calculated, for example, by Geogebra's probability calculation with a program that selects a binomial distribution as the probability distribution.

GeoGebra:

n = number of repetitions

p = Probability of a favorable event. In this case, the probability of a single grain of rice hitting a square

P (15 ≤ X ≤ 30) = Total probability that 15-30 rice grains will hit a white square.

Picture from Geogebra:

Answer: The probability that at least 15 rice grains will hit white is approximately 0,031

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