# 6 - Probability

When you throw a dice, you don't know on which side it will fall. You've got "one chance in 6 to get a 2", "one chance in 6 to get a 1" or "3 chances in 6 to get an odd number".

## Vocabulary

In the probability language, to throw a dice is random experiment. You can't foresee the result. Tomorrow's weather is a random experiment too because nobody, even the weather forecast, can really say what the weather will be tomorrow. The different results of a random experiment are called elementary events. An event is a sum of elementary events. The universal set is the set of elementary events associated to a random experiment. We denote it by . Probability theory is very closely associated to set theory, then when you throw a dice, the results {1}, {2}, {3}, {4}, {5}, {6} are each one elementary events, is an event and the universal set is the set .

## Formula

If all results have the same chance to happen, the elementary events are equally likely outcomes. It's the case when you throw a dice. In that case, if A is an event, then the probability of the event A is a number between 0 and 1 which is :

For example if A is the event 'to get a number greater or equal to 3', then

The elementary events are not always equally likely. If today is a sunny day, the probability that tomorrow will be sunny is not same that the probability that it will snow tomorrow.

## The deck of cards

Luck, Michael, Barack and Steeve are playing tarot. In a tarot deck there are 78 cards, 22 of these are trumps and there are 14 cards of each color. Luck icks up a card in the deck. What is the probability for the card to be a 5 but not the 5 of trump ? Thanks to the last formula, this probability is

Michael picks up a card. What is the probability for the card to be smaller than 5? There are 4 of each color, then

The same way, the probability for Barack to pick up a trump is .
Then Steeve picks up a card. What is the probability for the card to be a 2 or a heart ? The answer is not all the hearts added to all the 2's because the 2 of hearts would counted twice ! The solution is

Because we count twice the 2 of hearts, we subtract it once. To get 3 different probabilities, we can also write :

Usually, if A and B are events, we denote by (this reads 'A inter B' or 'A cap B') the sets of elementary events that are both in A and B, and we denote by ('A union B' or 'A cup B') the set of elementary events that are in either in A or in B (or both). You can translate Steeve's experiment by :

If the events A and B don't contain any common elementary event, we say that they are independent (for example with the dice if A = {1 ; 2} and B = {5 ; 6} ). Then , thus (the probability of an impossible event is null). The formula below is :

To understand this chapter, you can also represent the events with circles.

If two circles intersect then the area of the whole surface is equal to the area of the first circle added to the area of the second circle minus the area of their intersection. If the two circles don't intersect then you just have to add the areas.

## Mean value or average

A random variable is an unknown number whose value depends the result of a random experiment. For example, if we play the following game : "you stake one dollar, you throw a dice, if the number is smaller than 5 you lose the stake, if you obtain 5 you earn 2 dollars, if you obtain 6 you earn 3 dollars", we can say that the money earned at the end is an random variable that we call X. X can take the values -1, 2, or 3. The mean (or average) of X, denoted E(X) is what you can expect to earn by playing this game once. You calculate E(X) by multiplying all the possible earnings with their probabilities.

>>> Lesson on barycenters >>>

Probability

lesson, problems