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It is one of the simplest probability laws, Bernoulli's law is essential to know when you start in probability!

## Definition

Bernoulli's law with parameter p designates a discrete probability law which takes the value 1 with the probability p and 0 with the probability 1-p. It is therefore defined on the universe Ω = {0,1}.

Very concretely, you throw a coin (unbalanced) which has a probability p of heads, which we will note 1 and 0 of tails. Then coping follows a Bernoulli law.

A true/false situation that can be modeled by Bernoulli's law is called Bernoulli's test.

Bernoulli's law with parameter p is denoted B(p)

## Manufacturing

### Bernoulli's law expectation

The expectation of Bernoulli's law is** p. **It is very easy to demonstrate. Let X be a random variable

\begin{array}{ll} \mathbb{E}(X) &= \mathbb{P}(X=0) \times 0+ \mathbb{P}(X=1) \times 1 \\ &= ( 1-p) \times 0 + 1 \times p \\ &= p \end{array}

### Variance of Bernoulli's law

The variance of Bernoulli's law is** p(1-p). **It is quite easy to demonstrate. Let X be a random variable, we have:

\begin{array}{ll} \mathbb{E}(X^2 ) & = \mathbb{P}(X=0) \times 0^2+ \mathbb{P}(X=1) \times 1^ 2 \\ &= (1-p) \times 0 + 1 \times p \\ &= p \end{array}

We then have:

\begin{array}{ll} \mathbb{V}(X) &= \mathbb{E}(X^2) - \mathbb{E}(X)^2 \\ &= p - p^2 \\ &= p(1-p) \end{array}

## Corrected Bernoulli's law exercises

### Exercise 1

States : We have a deck of 52 cards. Determine the probability of drawing the following events:

- The one tile card
- The card is a king
- The map is a figure

Question 1 :

We note X the random variable which is worth 1 if we have a square and 0 otherwise. We have 4 different colors. The probability of this Bernoulli test is therefore

\mathbb{P}(X=1)=\dfrac{1}{4}

Question 2 :

There are 13 different values on the cards: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king. So we have a probability of

\dfrac{1}{13}

to have a king.

Question 3 :

There are 3 possible figures among the different values available on a card. So we have a probability of

\dfrac{3}{13}

to have a head.

### Exercise 2

States : Consider an urn containing 5 red balls and 6 white balls. Consider Bernoulli's test “we draw a ball from the urn”. This test will be a success if the drawn ball is white. What is the probability that the test fails?

Corrected : The test fails if a red ball is drawn. We have 5 red balls and a total of 11 balls. The probability that the test fails is therefore

\dfrac{5}{11}

### Exercise 3

States : Given a Bernoulli test with parameter

p = \dfrac{1}{4}

Calculate its expectation and its variance

Corrected :

His expectation is p. So his expectation is

\dfrac{1}{4}

We have :

1-p= \dfrac{3}{4}

We can therefore calculate the variance:

\mathbb{V}(X) = p(1-p) = \dfrac{1}{4}\dfrac{3}{4}=\dfrac{3}{16}

## Exercise statements

### Exercise 1

For each of the following tests, indicate whether it is a Bernoulli test and specify the success and its probability if applicable.

- We shoot a queen of hearts
- We look at the color of the card
- We check if the value is between 5 and 10 (inclusive)
- We check that the color of the card is red

### Exercise 2

Consider an urn containing 8 blue balls and 12 green balls. Consider Bernoulli's test “we draw a ball from the urn”. This test will be a success if the drawn ball is blue. What is the probability that the test fails?

### Exercise 3

An airplane has two identical engines. The probability of each failing is 0,005. It is assumed that the failure of one motor has no influence on the failure of the other motor. Calculate the probability that the plane will crash using a weighted tree.

### Exercise 4

Let X be a random variable following a Bernoulli distribution with parameter p. Knowing that the variance of X is equal to

\dfrac{6}{25}

Calculate the possible values of p.

### Exercise 5

Jean-Claude has four loyalty cards from separate stores in his pocket. These five cards all have the same format and are obviously indistinguishable to the touch.

At checkout in one of these stores, he chooses a loyalty card at random. Justify that this random experiment corresponds to a Bernoulli test by specifying the success and the probability of it.