Probability of two events · Years 9–10
In Probability of two events (Years 9–10) you practise 5 skills: Probability basics · Events in two stages in a table · With and without replacement · The area model · Probability and expected frequency. In an event in two stages, such as spinning two spinners, each outcome is a pair: what came up first and what came up second. A table with rows for the first stage and columns for the second shows every outcome. When all the outcomes are equally likely, the probability is the number of matching cells over the number of all the cells.
When two balls are taken from a bag, multiply the probabilities of the two stages. With replacement, the second stage is the same as the first. Without replacement, the bag has one ball fewer in the second stage. To find “at least one”, it is easier to work out the complement and subtract it from 1. The expected frequency is the probability times the number of trials. The other way round, the relative frequency in an experiment estimates a probability that cannot be worked out.
Tip: A probability is always between 0 and 1. If you get more than 1, there is a mistake.
What you learn
Probability basics
For example: A bag contains 9 red balls and 2 blue balls. One ball is taken out without looking. What is the probability of taking out a red ball?
Answer and solution
The answer: 9/11
There are 11 balls in the bag, and 9 of them are red. The probability is: 9/11
Events in two stages in a table
For example:
| First / second | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 |
| 2 | 3 | 4 | 5 | 6 |
| 3 | 4 | 5 | 6 | 7 |
Answer and solution
The answer: 1/6
The number of all the outcomes: 3 × 4 = 12 . The ones you want: (2, 4) , (3, 3) : 2 outcomes. The probability: 2/12 = 1/6
With and without replacement
For example: A bag has 2 red balls and 6 blue balls. A ball is taken out, its colour noted, and it is put back. Then a second ball is taken out. What is the probability that both balls are red?
Answer and solution
The answer: 1/16
First stage: of the 8 balls, 2 are red, so the probability of a red ball is 2/8 . In the second stage the ball is back, so the probability is again 2/8 . Multiply along the branch: 2/8 × 2/8 = 4/64 = 1/16
The area model
For example: A dart lands at random on the board, with the same chance of hitting any point. What is the probability that it hits the shaded region?
Answer and solution
The answer: 1/4
The shaded cells: 4 × 1 = 4 . On the whole board: 4 × 4 = 16 . The probability: 4/16 = 1/4
Probability and expected frequency
For example: An ordinary die is thrown 60 times. About how many times would you expect a number greater than 4?
Answer and solution
The answer: 20
The probability on one throw: 2/6 . The expected frequency: 2/6 × 60 = 20
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).