Probability
Probabilities can be written as fractions, decimals or percentages on a scale from 0 to 1. Knowing basic facts about equally likely outcomes can help to solve more complicated problems.
Probability and the probability scale
Probability is about estimating or calculating how likely or probable something is to happen. Probabilities can be described in words. For example, the chance of an event happening could be âcertainâ, âimpossibleâ or âlikelyâ.
In maths, probabilities are usually written as fractions, decimals or percentages with values between 0 and 1.
An event which is impossible has a probability of 0 and an event which is certain has a probability of 1. This means probabilities cannot be bigger than 1. This can be shown on a probability scale.
Other probability terms
As well as being familiar with the words, 'certain', 'evens' and 'impossible', there are some other terms that you need to know.
Likely and unlikely
The words 'likely' and 'unlikely' can be added to the probability scale.
If an event is likely, the likelihood of it happening is between 'evens' and 'certain'.
If an event is unlikely, the likelihood of it happening is between 'Impossible' and 'evens'.
Random
Choosing an item at random means that all the items have an equal chance of being chosen. Probability questions often say something like 'a disc is chosen at random'.
Fair
The word 'fair' is often used to describe a dice. It means that all six numbers are equally likely to show up when the dice is rolled.
Biased and unbiased
If a dice or spinner is described as biased, some outcomes are more likely than others.
Unbiased means that all outcomes are equally likely.
The word 'unbiased' is often used instead of fair.
Question
Match the events described below with the letters on the probability scale.
It will snow in June next year in Northern Ireland
You will get an even number when you roll a dice
Your teacher's birthday will be on a weekday next year
This year, Christmas Day will be on December 25th
Listing outcomes
In probability terms, the outcome is what happens when a coin is tossed, a dice is rolled or a spinner is spun etc.
Example
The possible outcomes when two dice are rolled and the numbers are added are:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12
Question
List all the different ways to get a total of 5 when two dice are rolled.
Answer
| First dice | Second dice |
|---|---|
| 2 | 3 |
| 3 | 2 |
| 4 | 1 |
| 1 | 4 |
There are four different ways to get a total of 5.
Probability as a fraction or decimal
For some situations, it is possible to quantify probability instead of using words.
Probability can be expressed as a number between 0 and 1 and may be written as a fraction or a decimal.
\(\text{probability of an outcome} = \frac{\text{number of ways the outcome can happen}}{\text{total number of possible outcomes}}\)
Example
There are 4 blue discs and 5 red discs in a bag. A disc is selected at random.
What is the probability that a red disc is selected?
Solution
\(\text{probability of an outcome} = \frac{\text{number of ways the outcome can happen}}{\text{total number of possible outcomes}}\)
There are 5 red discs and 9 discs in total.
Probability of a red disc being selected = \(\frac{5}{9}\)
Question
A fair dice is rolled. What is the probability of getting a number less than 3?
\(\text{probability of an outcome} = \frac{\text{number of ways the outcome can happen}}{\text{total number of possible outcomes}}\)
There are 2 ways to get a number less than 3 (they are 1 and 2).
There are 6 possible outcomes from rolling a dice.
Probability of a number less than 3 \(=\frac{2}{6}\)
\(\frac{2}{6}\) cancels down to give \(\frac{1}{3}\) but there is no need to simplify fractions when doing probability questions unless you have been asked to give your answer in its simplest terms.
Probability of events not happening
Events that cannot happen at the same time are called mutually exclusive events. For example, a football team can win, lose or draw but these things cannot happen at the same time - they are mutually exclusive. Since it is certain that one of these outcomes will happen, their probabilities must add up to 1.
If the probability the team wins is 0.5 and the probability it draws is 0.2 then the probability of it losing must be 0.3.
The probability of an event not happening is 1 minus the probability of the event happening.
Example
A bag contains 12 counters of different colours: 5 red, 4 white and 3 black. Find the probability of not selecting a red counter.
Solution
The probability of selecting a red counter is \(\frac{5}{12}\), so the probability of not selecting a red counter is\(1 â \frac{5}{12}\) which is \(\frac{12}{12} â \frac{5}{12} = \frac{7}{12}\).
Question
The probabilities of a spinner landing on a number are listed below. Find the probability \(p\) of the spinner landing on a 4.
| Number on spinner | 1 | 2 | 3 | 4 |
| Probability | 0.5 | 0.2 | 0.12 | p |
Answer
Probabilities of events add up to 1, so to find the probability of the spinner landing on a 4, add up the remaining probabilities and subtract this from 1.
\(0.5 + 0.2 + 0.12 = 0.82\)
\(p = 1 â 0.82 = 0.18\), so the probability of the spinner landing on a 4 must be 0.18.
Question
The probability of Anna being late when she misses her usual bus is 0.6.
What is the probability of Anna being on time when she misses this bus?
Answer
\(P (\text{Anna being on time}) = 1 â P (\text{Anna being late})\)
\(= 1 â 0.6 = 0.4\)
Expectation
Example
A fair dice is rolled 300 times.
How many times would you expect to roll a 5?
Solution
The probability of rolling a 5 with a fair dice is \(\frac{1}{6}\).
Expected number of successes \(= (\text{probability of a success}) \times (\text{number of trials})\)
Expected number of 5s \(= \frac{1}{6} \times 300 = 50\)
Question
The probability of tossing a head on a biased coin is \(\frac{2}{3}\).
If this coin is tossed 60 times, how many heads would you expect to get?
Answer
\(\frac{2}{3} \times 60 = 40\)
You would expect to get 40 heads.
Question
A four-sided spinner is spun 120 times.
How many times would you expect the spinner to land on a 3?
\(\text{expected number of successes} = P (\text{success}) \times \text{number of trials} \)
- Success in this question is getting a 3 on the spinner.
\(P (\text{getting a 3}) = \frac{1}{4}\)
Expected number of 3s \(= \frac{1}{4} \times 120\)
\(= 30\)