$$\mathrm{probability}=\frac{\mathrm{number\; of\; successful\; outcomes}}{\mathrm{total\; number\; of\; possible\; outcomes}}$$

$$P(X=x)=\frac{N(X=x)}{n}$$

As $0\le N(X=x)\le n$ then $0\le P(X=x)\le 1$.

If the probability of an event is $P$ then the probability of it not happening is $1-P$

If all outcomes are includes in the list of events then the list is exhaustive.

The probabilities of an exhaustive list of mutually exclusive events sum to one.

$$P(X=x)$$ This is read as the probability, $P$ of an event, $X$ is equal to an outcome, $x$ from the sample space.

Predicted number of outcomes = probability × number of trials $$n(X=x)=NP(X=x)$$

A sample space diagram shows all the possible outcomes. You can use it to find the theoretical probability.

You can estimate the probability of an event from the results of an experiment or survey $$\mathrm{estimated\; probability}=\frac{\mathrm{frequency\; of\; an\; event}}{\mathrm{total\; frequency}}$$

### The Addition Rule

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:

$$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$$Don't panic, this just means: The probability of $A$ or $B$ occurring is the probability of $A$ add the probability of $B$ minus the probability that they both occur. This is best seen by an example...

### Example:

Pick a card at random from a pack of 52 cards. Find the probability that you pick an ace or a spade. We want P(ace or spade) or P(ace ∪ spade). P(ace ∪ spade) = P(ace) + P(spade) − P(ace ∩ spade) = 4/52 + 13/52 − 1/52 = 16/52 or 4/13

### Mutually Exculsive Events:

Mutually exclusive events are those that cannot happen at the same time.

For
**mutually exclusive** events,
on a Venn diagram ther is no intersection, so
$P(A\cap B)=0$
and the addtion rule becomes
$P(AorB)=P\left(A\right)+P\left(B\right)$;

### Remember:

```
$P(\mathrm{ace}\cap \mathrm{spade})=P(\mathrm{ace}and\mathrm{spade})$
P(ace ∩ spade) = P(ace and spade)
= 4/52 × 13/52
= 1/52 i.e. the card is the ace of spades
```

- Probabilities and frequencies of events can be shown using a
**Venn diagram**; -
For
**independent**events, $P(AandB)=P\left(A\right)\times P\left(B\right)$; - Two or more events can also be represented by a
**tree diagram**.

(taken from Edexcel A Level year one syllabus)

- Probability Websites:
- - Probability Tree Diagrams.
- - Probability mass function;
- - Discrete Random Variables;
- - Pascal's Triangle;
- - Quincunx;