- In general this means there is more than one event to bear in mind when considering probabilities
- these events may be
**independent**or**mutually****exclusive** - they may involve an event that follows on from a previous event
- e.g. Rolling a dice, followed by flipping a coin

- these events may be

- In your head, try to rephrase each question as an AND and/or OR probability statement
- e.g. The probability of rolling a 6 followed by flipping heads would be "the probability of rolling a 6
**AND**the probability of flipping heads" - In general,
- AND means multiply (
) and is used for
**independent**events - OR mean add (
) and is used for
**mutually exclusive**events

- AND means multiply (
) and is used for

- e.g. The probability of rolling a 6 followed by flipping heads would be "the probability of rolling a 6
- The fact that all probabilities sum to 1 is often used in combined probability questions
- In particular when we are interested in an event "happening" or "not happening"
- e.g. so

- In particular when we are interested in an event "happening" or "not happening"
- Tree diagrams can be useful for calculating combined probabilities
- especially when there is more than one event but you are only concerned with two outcomes from each
- e.g. The probability of being stopped at one set of traffic lights and also being stopped at a second set of lights

- however unless a question specifically tells you to, you don't have to draw a diagram
- for many questions it is quicker simply to consider the possible options and apply the AND and OR rules without drawing a diagram

- especially when there is more than one event but you are only concerned with two outcomes from each

A box contains 3 blue counters and 8 red counters.

A counter is taken at random and its colour noted.

The counter is put back into the box.

A second counter is then taken at random, and its colour noted.

Work out the probability that

i)

both counters are red,

ii)

the two counters are different colours.

i)

This is an **"AND"** question: 1st counter red **AND** 2nd counter red.

ii)

This is an **"AND"** and **"OR"** question: [ 1st red **AND** 2nd green ] **OR** [ 1st green **AND** 2nd red ].

In the second line of working in part (ii) we are multiplying the same two fractions together twice, just 'the other way round'.

It would be possible to write that instead as

That sort of 'shortcut' is often possible in questions like this.

The probability of winning a fairground game is known to be 26%.

If the game is played 4 times find the probability that there is **at least one** win.

Write down an assumption you have made.

At least one win is the opposite to no losses so use the fact that the sum of all probabilities is 1.

Use the same fact to work out the probability of a loss.

The probability of four losses is an "AND" statement; lose AND lose AND lose AND lose.

Assuming the probability of losing doesn't change, this is
.

**P(at least 1 win) = 0.7001 (4 d.p.)**

The assumption that we made was that the probability of winning/losing doesn't change between games.

Mathematically this is described as each game being **independent.**

I.e., the outcome of one game does not affect the outcome of the next (or any other) game.

**It has been assumed that the outcome of each game is independent.**

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