Here I will explain how to calculate the probability of an isolated event with Laplace’s Law and also how to calculate the probability of the union of two events when they are compatible and when they are incompatible.

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## Laplace Law

Laplace’s Law is nothing more than the formula for calculating the probability of an isolated event occurring.

When in a random experiment, all events have the same probability of occurring, the probability of an A event occurring is:

In the numerator we place the number of favorable cases for event A to occur and in the denominator we place the number of possible events.

For example, on a die, what is the probability of obtaining a 2?

In this case, the number of favorable cases is 1, since the die has only one 2. The number of possible cases is 6, which are the numbers that a die has.

Therefore, the probability of a 2 coming out can be written as P(2) and is equal to:

The value of the probability of occurrence of event A shall be between 0 and 1:

With 1 being the probability of the safe event (it will always occur) and 0 being the probability of the impossible event (it will never occur).

## How to calculate the probability of occurrence of A or B, if the events are incompatible

If A and B are two incompatible events, i.e. they cannot occur at the same time, the probability of A occurring or B occurring will be the sum of the probabilities of each event occurring separately.

P(A ∪ B) reads: the probability of A or B occurring.

For example, in a bag we have 4 balls: one black, one white, one blue and one red. You have to extract a ball, what is the probability that it is white or black?

The event “take a cue ball” and the event “take a black ball” are two incompatible events because we can not take a cue ball and a black ball at the same time, therefore, the probability of taking a ball and that it is white or black will be the sum of each probability separately:

The probability of getting a cue ball is:

The probability of getting a black ball is:

Therefore, the probability of getting a cue ball or black ball is:

## How to calculate the probability of A or B occurring, if the events are compatible

If A and B are two compatible events, that is, they can occur at the same time, then the probability of A or B occurring will be:

P(A ∩ B) reads: the probability of A and B occurring.

This time, the sum of the probabilities of each event occurring separately must be subtracted from the probability of the two events occurring at the same time.

For example, calculate the probability that when you roll a die, the number obtained is even or that it is a 4.

In this case the event “get even number” and the event “get a 4” are compatible because if we get a 4 are happening both events at once.

Therefore, the probability of getting 4 or an even number will be calculated with the formula:

The probability of getting an even number is:

The probability of getting a 4 is:

The probability of getting an even number and a 4 is:

There is only one possibility between 6 since 4 is the only number that fulfills both events at the same time.

Finally, the probability of getting an even number or 4 is:

That substituting each term for its value is left to us:

## Proposed exercises

1- Some cards have been lost from a Spanish deck of 40 cards. If we draw a card among the ones we have left, the probability that it is of clubs is 0.33, the probability that it is a king is 0.14 and that it is of clubs or king is 0.44. Is the king of clubs among the cards we have? *Suggestion: Calculate the probability of removing the king of clubs (B∩R)*.

2- In one coin purse I have 3 coins of 1 euro and 7 of 50 cents. In another wallet I have 4 1 euro and 4 50 cent coins. If I randomly draw a coin from each purse, how likely is it that I will get 1.50 euros?

**Exercise 1**

The statement tells us that the probability of a chart being rough is 0.33:

The probability that it is a king is:

The probability that he is a king or that he is a bastos is:

As the events “to be of clubs” and “to be a king” are compatible events, since the king can be of clubs, the probability of drawing a card that is a king or of clubs can be calculated with the formula:

From where we know all the terms except the probability that it is coarse and king, that is, that it is the king of coarse.

We replace each probability with its value:

Y from here we clear the probability that it is coarse and king:

That is greater than zero, therefore, there is a chance that the king of clubs is in the deck.

**Exercise 2**

To get €1.50 you need to draw a €1 coin from one purse and another €0.50 coin from the other purse.

The probability of withdrawing a 1 € coin from the first purse is:

The probability of withdrawing a €0.50 coin in the first purse is:

The probability of withdrawing a 1 € coin in the second purse is:

The probability of withdrawing a coin of 0,50 € in the second purse is:

Removing a coin from one purse and then removing the coin from another purse are dependent events and because one event depends on another. In this case, the probabilities multiply.

We can start by removing the coin from the first purse:

The probability of withdrawing 1 € from the first purse and then withdrawing 0.50 € from the second purse is:

Or we can start by removing the coin from the second purse:

The probability of withdrawing 1 € from the second coin purse and then withdrawing 0,50 € from the first coin purse is:

The probability of getting 1,50 € will be the probability of starting with the first purse plus the probability of starting with the second purse, as they are incompatible events (either you start with one or you start with another, but not both at the same time):

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