# 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 npr.

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.

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, 