## Laws of probability (A level Statistics)

## Mutually exclusive events and the addition law

**Mutually exclusive events** are events that cannot occur at the same time.

Let us say we have a class of 10 students and we want to random remove 1 student from the class. The student should be either a boy **or** a person who lives close to the school. In this situation we have two events.

If we choose a boy, then we wont have to choose a person who lives close to the school, and if we choose a person who lives close to the school, then we wont have to choose the boy, because the condition is guided by the keyword “or”.

For mutually exclusive events we use the addition law:

P(A or B) = P(A) + P(B)

In probability, the addition law is recognized by the keyword “or” joining the probabilities of the mentioned events.

## Independent events and the multiplication law

An **independent** event is an event whose probability of happening does not affect the probability of another event happening.

In probability, the multiplication law is recognized by the keyword “and” joining the probabilities.

For independent events we use the multiplication law:

P(A and B) = P(A) × P(B)

## Probability worked examples

### Example 1

A group of 20 athletes competed in an athletic event in which only one athletic will be crown the winner. Calculate the probabilities that:

a) one athlete is chosen before the event at random is going to win the event.

b) of the two athletes chosen at random before two consecutive events, one is going to win the first event and the other is going to win the next event.

### Solution 1

a) P(winner) = $latex \frac{1}{20}$

b) P(winner and winner) = P(winner) × P(winner)

= $latex \frac{1}{20}$ × $latex \frac{1}{20}$

= $latex \frac{1}{400}$

### Example 2

It is given that during a certain rocket test launch, the probability of the main engine failing due to excessive temperature is $latex \frac{1}{100}$, due to excessive vibration is $latex \frac{1}{50}$ and due to undetected manufacturing defects is $latex \frac{1}{150}$.

Determine the probabilities that in one test launch the main engine will:

a) fail due to both excessive temperature and excessive vibration

b) fail due to excessive vibration or excessive temperature

c) not fail due to both excessive temperature and not due to an undetected manufacturing defect.

### Solution 2

Let *A* be the event where the main engine fails due to excessive temperature.

∴ P(A) = $latex \frac{1}{100}$.

and the probability of the main engine not failing due to excessive temperature becomes:

P(not A) = 1 – P(A)

= $latex \frac{99}{100}$.

Let *B* be the event where the main engine fails due to excessive vibration.

∴ P(B) = $latex \frac{1}{50}$

and the probability of the main engine not failing due to excessive vibration becomes:

P(not B) = 1 – P(B)

= $latex \frac{49}{50}$.

Let *C* be the event where the main engine fails due to undetected manufacturing defects.

∴ P(C) = $latex \frac{1}{150}$.

and the probability of the main engine not failing due to undetected manufacturing defect becomes:

P(not C) = 1 – P(C)

= $latex \frac{149}{150}$.

a) P(A and B) = P(A) × P(B)

= $latex \frac{1}{100}$ × $latex \frac{1}{50}$

= $latex \frac{1}{5000}$

b) P(A or B) = P(A) + P(B)

= $latex \frac{1}{100}$ + $latex \frac{1}{50}$

= $latex \frac{3}{100}$

c) P(not A and not C) = $latex \frac{99}{100}$ × $latex \frac{149}{150}$

= $latex \frac{4917}{5000}$