Probability of combined events - KS3 Maths

Key points

A series of four images. Each image shows a pair of bags with two counters. In the first bag there is one blue and one purple counter. In the second bag there is one purple and one green counter. The first image represents a blue counter selected from the first bag and a purple counter selected from the second bag. Written right: blue, purple. The second image represents a blue counter selected from the first bag and a green counter selected from the second bag. Written right: blue, green. The third image represents a purple counter selected from the first bag and a purple counter selected from the second bag. Written right: purple, purple. The fourth image represents a purple counter selected from the first bag and a green counter selected from the second bag. Written right: purple, green. Written below: outcomes, B P, comma, B G, comma, P P, comma, P G. The word, blue, and the letter, B, are coloured blue. The word, purple, and the letter, P, are coloured purple. The word, green, and the letter G, are coloured green. Each image has the correct coloured counters highlighted in the first and second bags. — Image caption,
The possible outcomes, BP, BG, PP, PG, can be used to work out probabilities.

involve two or more separate . Rolling a die and flipping a coin or rolling two dice produce combined events.
The can be used to find the total number of outcomes of combined events.
The of combined events may be listed or written in the form of a table called a . These can be used to find the probability of a particular outcome.
The is usually written as a fraction. The total number of possible outcomes is the denominator. The number of times the outcome is possible is the numerator.

Listing outcomes to find the probability of combined events

Taking a logical and organised approach will make sure that all the outcomes are found. Just using the first letter of words, eg B for ‘blue’ can make listing faster.
To work out the total number of outcomes use the product rule:
- Multiply the number of outcomes for each event.
To list all the outcomes for a pair of combined events:
1. Start with one event, write down one of its outcomes paired with each outcome of the second event.
2. For the same event, write down another of its outcomes paired with each outcome of the second event.
3. Repeat until all the outcomes of the first event have been paired with each outcome of the second event.
Where there are three or more events, write the list carefully to make sure all outcomes are listed.
Write the probability of the combined events as a fraction.
- The denominator is the total number of possible outcomes.
- The numerator is the number of times the particular outcome occurs.
- It may be possible to simplify the fraction.

Examples

Skip image gallery

Image caption,
Two bags contain counters. The first bag has one blue counter and one purple counter. The second bag has one purple counter and one green counter. One counter is taken from each bag. Find the total number of possible outcomes and list them.
Image caption,
To find the total number of outcomes use the product rule. Multiply the number of outcomes of each event. The first bag has two outcomes (blue or purple), the second bag has two outcomes (purple or green). 2 × 2 = 4. There will be four combined outcomes.
Image caption,
To list the outcomes, work in an organised way. Start with one event, the counter taken from the first bag. One outcome is to take a blue counter from the first bag. When the first counter is blue, the counter from the second bag may be purple or green. This gives the outcomes blue purple and blue green.
Image caption,
Taking a counter from the first bag has another outcome. A purple counter may be taken from the first bag. When the first counter is purple, the counter from the second bag may again be purple or green. This gives the outcomes purple purple and purple green.
Image caption,
There are four outcomes altogether. First letters of words can be used. There are four possible outcomes which are BP, BG, PP, PG. These can be used to work out probabilities.
Image caption,
A counter is taken at random from each bag. Find the probability that the counters will be different colours.
Image caption,
There are four possible outcomes, BP, BG, PP, PG. The denominator is the total number of possible outcomes (4). Three of the outcomes are different colours. The numerator is three. The probability that the counters will be different colours is ¾.

Questions

Question 1:
Two spinners are twirled to decide which two types of fruit should go into a lunch box. The first spinner represents raspberries and blueberries. The second spinner represents strawberries, cherries and apples.

Find the probability that both portions of fruit are berries.

A series of two images. Each image shows a circular spinner with a black arrow in various positions. The first spinner is split in to two equal sectors. One sector is decorated with a picture of raspberries, and one sector is decorated with a picture of blueberries. The second spinner is split in to three equal sectors. One sector is decorated with a picture of a strawberry, one sector is decorated with a picture of an apple, and one sector is decorated with a picture of cherries.

Question 2:
Three spinners are used to select a starter, main and dessert for a three-course meal.

Use the product rule to find the total number of possible three-course meals.

A series of three images. Each image shows a circular spinner with a black arrow in various positions. The first spinner is split in to three equal sectors. Each sector is labelled in orange; soup, garlic bread, salad. Written above: starter. The second spinner is split in to four equal sectors. Each sector is labelled in blue; steak, fish, veg, pizza. Written above: main. The third spinner is split in to two equal sectors. Each section of the sector is labelled in pink; chocolate brownie, ice cream. Written above: dessert.

Using sample space diagrams to find the probability of combined events

A diagram is a grid used to show all the of two . The dimensions of the grid can be multiplied to give the total number of outcomes (this is the product rule).
To draw a sample space diagram:
1. Draw a grid, with dimensions showing the number of outcomes for one , by the number of outcomes for the second event. The orientation of the grid makes no difference to the number of outcomes.
2. Label each side of the grid with the outcomes of each event. The cells of the grid are filled with the combined outcome. This may be the result of a calculation or written to represent a combination of events.
Use the sample space diagram to work out the probability of the combined event as a fraction.
- The denominator is the total number of possible outcomes.
- The numerator is the number of times the particular outcome occurs.
- It may be possible to simplify the fraction.

Examples

Skip image gallery

Image caption,
A fair blue spinner is labelled 1, 2, 3 and 4. A fair green spinner is labelled 1, 2 and 3. Each spinner is twirled, and the numbers added together. Use a sample space diagram to find all the outcomes.
Image caption,
Draw a grid labelled with the outcomes of the green and blue spinners and the calculation (addition) to be done. In this case either a 4 by 3 or 3 by 4 grid may be used. The total number of outcomes is 12. (3 × 4 = 12 and 4 × 3 = 12).
Image caption,
Each cell of the grid is filled with the combined outcomes, the addition of the number on the blue spinner by the number on the green spinner. The highlighted cell shows 5 as the answer to 2 on the blue spinner added to 3 on the green spinner.
Image caption,
There are 12 outcomes shown in the sample space diagram.
Image caption,
The sample space diagram shows the outcomes when a green and blue spinner have their scores added. Find the probability that the score is 5
Image caption,
There are twelve outcomes, the denominator of the fraction is 12. The score of 5 occurs three times. The numerator of the fraction is 3. The probability that the score is five is ³⁄₁₂. This can be simplified to ¼. P(5) = ¼.
Image caption,
A bag contains one purple counter and one blue counter. A counter is taken out of the bag, its colour noted. The counter is replaced in the bag and a second counter is taken out and its colour noted. Use a sample space diagram to find all the possible outcomes and find the probability that the counters will be different colours.
Image caption,
Draw a 2 by 2 grid, labelling each column and row with the different colours. This is to show that the first and second counter may be either purple or blue.
Image caption,
Fill in the cells with the possible combinations. B represents blue, P represents purple. There are four outcomes, BB, BP, PB, and PP. The sample space diagram can be used to work out the probability that the two counters are different colours.
Image caption,
There are four outcomes, the denominator of the fraction is 4. The counters are different colours in two of the outcomes. The numerator of the fraction is 2. The probability that the counters are different colours is ²⁄₄. This can be simplified to ½. P(different colours) = ½.

Question

Two bags contain counters. The first bag contains two blue and one white counter. The second bag contains one blue, one purple and one white counter.

The sample space diagram shows all the possible outcomes of taking one counter from each bag.

Use the sample space diagram to find the probability of taking two counters of the same colour.

An image of a sample space diagram. The sample space diagram is a grid with three columns and three rows. The labels for the rows are, blue, blue, white. The labels for the columns are, purple, white, blue. An addition symbol has been drawn outside the top left corner of the grid. Each cell of the sample space has been populated. From left to right the first row reads, B P, B W, B B. From left to right the second row reads, B P, B W, B B. From left to right the third row reads, W P, W W, W B. Written below: P, open bracket, same colour, close bracket, equals question mark. The question mark is coloured orange.

Probability activity

Play this game to explore probabilities. You're given a choice of four dice that don't all have standard faces. The numbers that appear on the dice are given in the table.

Try to find the combination of dice that gives you the highest probability of rolling a total of 7.

Practise working out probability of combined events

Practise working out probability of combined events with this quiz. You may need a pen and paper to help you with your answers.

Quiz

Real-life maths

An image of a special sample space diagram called a Punnett square. The sample space diagram is a grid with two columns and two rows. The labels for the rows are, Capital C, lower case c. Written left: female parent. The labels for the columns are, Capital C, lower case c. Written above male parent. Each cell of the sample space has been populated. From left to right the first row reads, Capital C Capital C, Capital C lower case c. From left to right the second row reads, Capital C lower case c, lower case c lower case c. Written beneath, the bullet points: Capital C Capital C, No cystic fibrosis, probability twenty five percent. Capital C lower case c, cystic fibrosis carrier, probability fifty percent. Lower case c lower case c, has cystic fibrosis, probability twenty five percent. Without cystic fibrosis, seventy five percent. With cystic fibrosis, twenty five percent. The labels for the rows are highlighted purple. The labels for the columns are highlighted orange. — Image caption,
Punnett squares can be used to work out probabilities, like a sample space diagram.

A is an inherited medical condition. It can be passed from parents to their children. Examples include cystic fibrosis, sickle cell disease and haemophilia.

Doctors can carry out tests to identify whether a baby has an inherited condition if a parent is affected by it or knows they are a .

These conditions are passed on to children through the that make up the of their parents. For example, cystic fibrosis can be inherited if both parents carry the gene, even though they do not suffer from cystic fibrosis themselves. use Punnett squares to work out the probability of something being inherited. They are like a sample space diagram used for working out probabilities of combined events.

In the Punnett square above, C represents the gene without cystic fibrosis and c is the gene for cystic fibrosis. These are the potential outcomes for two parents that are carriers of the cystic fibrosis genes and their probability of having a child with cystic fibrosis.