Probability

Probability is about estimating the chance or how likely (probable) something is to happen. For example, an outcome when throwing a die or tossing a coin.

We often use words to describe how probable we think it is that an event will take place.

For example, we might say that it is likely to be sunny tomorrow, or that it is unlikely to snow in August.

Have a look at the statements below, and decide whether the following events are:

• certain
• likely
• even (neither likely or unlikely)
• unlikely
• impossible

Question
1. You buy a lottery ticket and win the jackpot
2. You toss a coin and get heads
3. Christmas will fall on 25 December this year
4. You grow another nose
5. It will rain in the first week of December
1. You buy a lottery ticket and win the jackpot - Unlikely
2. You toss a coin and get heads - Even chance
3. Christmas will fall on 25 December this year - Certain
4. You grow another nose - Impossible
5. It will rain in the first week of December - Likely

Words like 'certain', 'likely', and so on, may not mean the same to everyone.

We need to be more precise about how likely something (an outcome) is to happen.

The probability of an outcome can have any value between 0 (impossible) and 1 (certain).

It may be written as a fraction, decimal fraction or percentage.

We can calculate a value for the probability using the formula:

$probability\,of\,an\,outcome = \frac{{number\,of\,ways\,the\,outcome\,can\,happen}}{{total\,number\,of\,possible\,outcomes}}$

The table below gives some examples of events and how their probability can be calculated.

EventOutcomeNumber of ways to get this outcomeTotal number of possible outcomesProbability of outcome
Throwing a fair, 6-sided diceGetting an odd number36$\frac{3}{6} = \frac{1}{2}$
Throwing a fair coinGetting 'tails'12$50\%$
Choosing a playing card from a full pack without lookingThe suit being spades1352$\frac{13}{52} = \frac{1}{4}$
Choosing a playing card from a full pack without lookingThe card being a 'ten'452$\frac{4}{52} = \frac{1}{13}$
Throwing a fair, 6-sided diceGetting a number less than 546$\frac{4}{6} = \frac{2}{3}$

Example

Imagine there are two fair spinners.

A triangular one is marked with one number on each side: 1, 2 and 3.

A square one is also marked with one number on each side: 0, 1, 2 and 3.

The total score is the sum of the two numbers the spinners land on.

Jot down, systematically, all the possible outcomes for the two spinners. You will find it useful to use a table of results, as shown.

Triangular spinnerSquare spinnerTotal score
101
112
123
134
202
213
224
235
303
314
325
336

Use the table to answer these questions.

Question

How many different possible outcomes are there?

There are 12 different possible outcomes.

All you have to do is count the number of rows in the table.

Question

How many outcomes give a total score of 2?

Only 2 outcomes give a total score of 2.

These are 1 + 1 and 2 + 0

Question

What is the probability of getting a total score of 2?

The number of outcomes giving a total score of 2 over the number of outcomes possible $$= \frac{2}{{12}}\,or\,\frac{1}{6}$$

Question

How many outcomes give a total score of 4?

3 outcomes give a total score of 4.

Question

What is the probability of getting a total score of 4?

The number of outcomes giving a total score of 4 over the number of outcomes possible $$= \frac{3}{{12}}\,or\,\frac{1}{4}$$

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