Finding probability
\(\text{The probability of an outcome} = \frac{\text{the number of ways the outcome can happen}}{\text{total number of possible outcomes}}\)
Think about a fair ordinary dice. To find the probabilityThe extent to which something is likely to be the case. of rolling a 4, take the number of possible ways of rolling a 4 and divide it by the total number of possible outcomes.
There is one way of rolling a 4 and there are six possible outcomes, so the probability of rolling a 4 on a dice is \(\frac{1}{6}\). This is called the ‘theoretical probability’ - in theory, if you roll a dice six times then you should roll a 4 once.
To find the probability of rolling an odd number on a dice, take the number of ways of getting an odd number which is 3 (1, 3 and 5), and divide by the total number of possible outcomes:
\(\frac{3}{6} = \frac{1}{2}\)
Question
What is the probability of selecting a vowel at random from the word PROBABILITY?
There are four vowels out of a total of eleven letters: PROBABILITY. This means the probability of selecting a vowel is \(\frac{4}{11}\).
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.
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.
The probability of selecting a red counter is \(\frac{5}{12}\), so the probability of not selecting a red counter is \(1 - \frac{5}{12}\) which is \(\frac{12}{12} - \frac{5}{12} = \frac{7}{12}\).
Question
A spinner has four sections numbered 1, 2, 3 and 4. The probabilities of the 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 | \(p\) |
| Probability | |
|---|---|
| 1 | 0.5 |
| 2 | 0.2 |
| 3 | 0.12 |
| 4 | \(p\) |
Probabilities of events add up to 1, so to find the probability of the spinner showing a 4, add up the remaining probabilities and subtract this from 1.
\(0.5 + 0.2 + 0.12 = 0.82\)
\(p = 1 - 0.82 = 0.18\), so the probability of the spinner showing a 4 must be 0.18.
Question
If the same spinner was spun 50 times how many times would you expect it to land on the number 2?
Since the probability of the spinner landing on a 2 is 0.2, the number of 2s expected from 50 spins would be \(50 \times 0.2 = 10\)
