# S-Cool Revision Summary

## S-Cool Revision Summary

#### Permutations

The number of permutations of arranging n distinct (different) objects is:

**n! (n factorial)**

**n! = n x (n - 1) x (n - 2) x ... x 2 x 1**

The number of ways of arranging n objects of which r are the same is:

In addition to this the number of ways of arranging **n** objects of **p** of one type are alike, **q** of a second type are alike, **r** of a third type are alike etc.

**This is given by: **

The number of permutations of **r** objects from **n** is written as ** ^{n}p_{r}**.

**We write: **

*Handy hint:* Nearly all permutation questions involve putting things in order from a line where the order matters. For example, ABC is a different permutation to ACB.

#### Combinations

Suppose that we wish to choose **r** objects from **n**, but the order in which the objects are arranged does not matter. Such a choice is called a combination. ABC would be the same combination as ACB as they include all the same letters.

**The number of combinations of r objects from n, distinct, objects can be written in 2 ways:**

#### Probability

The probability that an event, **A**, will happen is written as **P(A)**.

The probability that the event **A**, does not happen is called the **complement of A** and is written as **A'**

As either **A** must or must not happen then,

**P(A') = 1 - P(A)** as probability of a certainty is equal to 1.

#### Set notation

**If A and B are two events then:**

A B represents the event 'both A and B occur'

A B represents the event 'either A or B occur'

#### Mutually Exclusive Events

Two events are mutually exclusive if the event of one happening excludes the other from happening. In other words, they both cannot happen simultaneously.

**For exclusive events A and B then:**

P(A or B) = P(A) + P(B) this can be written in set notation as

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

This can be extended for three or more exclusive events

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

*Handy hint:* Exclusive events will involve the words 'or', 'either' or something which implies 'or'. **Remember 'OR' means 'add'.**

#### Independent Events

Two events are independent if the occurrence of one happening does not affect the occurrence of the other.

For independent events A and B then: **P(A and B) = P(A) + P(B)**

This can be written in set notation as: **P(A B) = P(A) + P(B)**

Again, this result can be extended for three or more events: **P(A and B and C) = P(A) + P(B) + P(C)**

*Handy hint:* Independent events will involve the words 'and', 'both' or something which implies either of these.

Remember 'and' means 'multiply'.

#### Tree diagrams

Most problems will involve a combination of exclusive and independent events. One of the best ways to answer these questions is to draw a tree diagram to cover all the arrangements.

#### The Addition Law

If two events, **A** and **B**, are not mutually exclusive then the probability that **A** or **B** will occur is given by the addition formula: **P(A B) = P(A) + P(B) - P(A B)**

The probability of **A** or **B** occurring is the probability of **A** add the probability of **B** minus the probability that they both occur.

#### Conditional Probability

If we need to find the probability of an event occurring given that another event has already occurred then we are dealing with Conditional probability.

If **A** and **B** are two events then the conditional probability that **A** occurs given that **B** already has is written as where:

Or:

If we rearrange this formula we obtain another useful result:

If the two events A and B are independent (for instance, one doesn't affect the other), then quite clearly,