Mutually exclusive and independent events

Here is a biased 5-sided spinner. When the spinner is spun, it can land on blue or on green or on purple or on red or on yellow.

The table gives information about the probability of the spinner landing on each colour.

Sophie spins the spinner once.

Work out the probability that the spinner lands on blue or on green or on purple.

Max spins the spinner 350 times.

Work out an estimate for the number of times the spinner lands on red.

Ricardo is going to play one game of badminton and one game of tennis.

He can either win or lose each game.

The probability that he will win the game of badminton is 0.7 The probability that he will win the game of tennis is 0.4

Complete the probability tree diagram.

badminton tennis

Work out the probability that Ricardo loses both games.

Kannika has 9 counters. There is a number on each counter.

Kannika puts the 9 counters in a bag. She takes at random a counter from the bag and does not replace the counter. She then takes at random a second counter from the bag.

Work out the probability that the sum of the numbers on the two counters is less than 5

A box contains 20 counters.

9 of the counters are red 7 of the counters are yellow 4 of the counters are green

Alex takes at random three counters from the box.

Work out the probability that exactly two of the three counters are the same colour.

Here is a biased spinner.

The table gives information about the probability that, when the spinner is spun once, it will land on each number.

Alexis is going to spin the spinner 400 times.

Work out an estimate for the number of times the spinner will land on an odd number.

Here is a biased spinner.

The table gives information about the probability that, when the spinner is spun once, it will land on each number.

Alexis is going to spin the spinner 400 times.

Work out an estimate for the number of times the spinner will land on an odd number.

Harman has two bags of beads.

In bag A, there are 3 white beads and 7 black beads. In bag B, there are 5 white beads and 4 black beads.

Harman takes at random a bead from bag A and a bead from bag B

Complete the probability tree diagram.

Bag A Bag B

Work out the probability that Harman takes two beads of the same colour.

There are 20 sweets in a box.

15 of the sweets are red 5 of the sweets are yellow

Fred takes at random 3 sweets from the box.

Work out the probability that Fred takes at least one sweet of each colour from the box.

There are 25 counters in a bag such that

6 counters are blue x counters are orange, where x > 9 the rest of the counters are pink

Maalam takes at random two of the counters from the bag. The probability that Maalam takes one orange counter and one pink counter is Calculate the probability that Maalam takes 2 pink counters from the bag. Show clear algebraic working.

Sid has 2 boxes of crayons, box X and box Y

5 of the 16 crayons in box X are red. 7 of the 20 crayons in box Y are red. Sid takes at random one crayon from box X and one crayon from box Y

Complete the probability tree diagram.

box X box Y

Work out the probability that Sid takes two crayons that are red or two crayons that are not red.

Novak is going to play two games of tennis.

The probability that he wins the first game is 0.7

If he wins the first game, the probability that he wins the second game is 0.9 If he loses the first game, the probability that he wins the second game is 0.4

Complete the probability tree diagram.

Work out the probability that Novak wins both games of tennis.

Narin has two fair 6-sided dice.

Dice A has 2 red faces and 4 yellow faces. Dice B has 1 red face and 5 yellow faces.

Narin is going to throw each dice once.

Complete the probability tree diagram.

Work out the probability that both dice land on yellow.

There are 12 counters in a bag.

3 of the counters are red 9 of the counters are green

Ameya, Jack and Ella each take at random one counter from the bag.

Work out the probability that at least one red counter is still in the bag.

Here are 9 cards. Each card has either a number on it or a letter on it.

Tomas is playing a game.

Tomas takes at random one of the cards and keeps it.

Tomas then takes at random another card and keeps it.

Complete the probability tree diagram.

First card Second card

Work out the probability that each of the two cards has a number on it.

Work out the probability that there will be one card with a number on it and one card with a letter on it.

Anjali wants to go on a boat at the seaside.

At the seaside there are 20 boats.

Of these boats 2 are white 5 are blue 7 are green 6 are yellow

Anjali selects at random one of these boats.

Write down the probability that she selects

a green boat,

a white boat or a yellow boat.

Here is a biased spinner.

When the spinner is spun once, the probabilities that it lands on red or on yellow or on green are given in the table.

Work out the probability that the spinner lands on red or on yellow.

Yang is going to spin the spinner 300 times.

Work out an estimate for the number of times the spinner will land on purple.

Rudolf goes to the gym.

The probability that he will use the treadmill is 0.8

When he uses the treadmill, the probability that he will use the cross trainer is 0.3 When he does not use the treadmill, the probability that he will use the cross trainer is 0.6

Complete the probability tree diagram for this information.

_........ use cross trainer

Work out the probability that Rudolf uses both the treadmill and the cross trainer.

Abraham is going to play a computer game. Abraham can win the game, draw the game or lose the game.

For any game that Abraham plays

the probability that he wins the game is 0.3 the probability that he draws the game is 0.5 the probability that he loses the game is 0.2

When Abraham wins a game, he scores +10 points. When Abraham draws a game, he scores 0 points. When Abraham loses a game, he scores −5 points.

Abraham plays 3 games and the points he scores in each of the 3 games are added together to get his total score.

Work out the probability that when he has played 3 games his total score is 0 points.

Elliot has x counters.

Each counter has one red face and one green face.

Elliot spreads all the counters out on a table and sees that the number of counters showing a red face is 5

Elliot then picks at random one of the counters and turns the counter over. He then picks at random a second counter and turns the counter over. The probability that there are still 5 counters showing a red face is

Work out the value of x Show clear algebraic working.

Ciara throws four fair six‑sided dice.

The faces of each dice are labelled with the numbers 1, 2, 3, 4, 5, 6

Work out the probability that at least one of the dice lands on an even number.

Aika has 2 packets of seeds, packet A and packet B

There are 12 seeds in packet A and 7 of these are sunflower seeds. There are 15 seeds in packet B and 8 of these are sunflower seeds.

Aika is going to take at random a seed from packet A and a seed from packet B

Complete the probability tree diagram.

Calculate the probability that Aika will take two sunflower seeds.

A tin contains tea bags with a choice of four different flavours of tea. The four flavours of tea are Assam or Darjeeling or Nilgiri or Rize.

Sara takes at random a tea bag from the tin.

The table shows each of the probabilities that the flavour of the tea Sara takes is Assam or Darjeeling or Rize.

Work out the probability that the flavour of the tea Sara takes is Nilgiri.

Work out the probability that the flavour of the tea Sara takes is either Darjeeling or Rize.

A tin contains tea bags with a choice of four different flavours of tea. The four flavours of tea are Assam or Darjeeling or Nilgiri or Rize.

Sara takes at random a tea bag from the tin.

The table shows each of the probabilities that the flavour of the tea Sara takes is Assam or Darjeeling or Rize.

Work out the probability that the flavour of the tea Sara takes is Nilgiri.

Work out the probability that the flavour of the tea Sara takes is either Darjeeling or Rize.

A group of 60 students each sat an algebra test and a geometry test. Each test was marked out of 110

The cumulative frequency graph gives information about the marks gained by the 60 students in the algebra test.

Use the graph to find an estimate for the median mark in the algebra test.

Use the graph to find an estimate for the number of students who gained 58 marks or less in the algebra test.

Use the graph to find an estimate for the interquartile range of the marks gained in the algebra test.

The interquartile range of the marks gained in the geometry test is 9

Luis says

“The students’ marks are more spread out in the algebra test than in the geometry test.”

Is Luis correct? Give a reason for your answer.

To be awarded a grade A in the algebra test, a student had to gain a mark greater than 64

Two students are to be selected at random from the 60 students in the group.

Use the graph to find an estimate for the probability that both of these students were awarded a grade A in the algebra test.

15Magnus and Garry play 2 games of chess against each other. The probability that Magnus beats Garry in any game is The probability that any game between Magnus and Garry is drawn is The result of any game is independent of the result of any other game.

Complete the probability tree diagram.

For each game of chess,

the winner gets 2 points and the loser gets 0 points, when the game is drawn, each player gets 1 point.

Work out the probability that, after 2 games, Magnus and Garry have the same number of points.

Magnus and Garry now play a third game of chess.

Work out the probability that, after 3 games, Magnus and Garry have the same number of points.

Ding is going to play one game of snooker against each of two of his friends, Marco and Judd.

The probability tree diagram gives information about the probabilities that Ding will win or lose each of these two games.

Ding against
Ding against

Work out the probability that Ding will win both games.

Work out the probability that Ding will win exactly one of the games.

A bag contains n beads. 6 of the beads are red and the rest are blue.

Ravi is going to take at random 2 beads from the bag. The probability that the 2 beads will be of the same colour is

Using algebra, and showing each stage of your working, calculate the value of n.

Harry has two fair 5-sided spinners.

Spinner A Spinner B

Harry is going to spin each spinner once.

Complete the probability tree diagram.

Work out the probability that at least one of the spinners will land on green.

Steffi is going to play one game of tennis and one game of chess.

The probability that she will win the game of tennis is 0.6 The probability that she will win both games is 0.42

Work out the probability that she will not win either game.

There are 16 sweets in a bowl.

4 of the sweets are blackcurrant. 5 of the sweets are lemon. 7 of the sweets are orange.

Anna, Ravi and Sam each take at random one sweet from the bowl.

Work out the probability that the 5 lemon sweets are still in the bowl.

Cody has two bags of counters, bag A and bag B.

Each of the counters has either an odd number or an even number written on it.

There are 10 counters in bag A and 7 of these counters have an odd number written on them. There are 12 counters in bag B and 7 of these counters have an odd number written on them.

Cody is going to take at random a counter from bag A and a counter from bag B.

Complete the probability tree diagram.

Calculate the probability that the total of the numbers on the two counters will be an odd number.

Harriet also has a bag of counters. Each of her counters also has either an odd number or an even number written on it.

Harriet is going to take at random a counter from her bag of counters.

The probability that the number on each of Cody’s two counters and the number on Harriet’s counter will all be even is

Find the least number of counters that Harriet has in her bag. Show your working clearly.

In group C, there are 6 girls and 8 boys. In group D, there are 3 girls and 7 boys.

A team is made by picking at random one child from group C and one child from group D.

Complete the probability tree diagram.

Work out the probability that there are two boys in the team.

After the first team has been picked, a second team is picked. One child is picked at random from the children left in group C and one child is picked at random from the children left in group D.

Work out the probability that there are two boys in each of the two teams.

Each day that Barney goes to college, he either goes by bus or he walks. The probability that Barney will go to college by bus on any day is 0.3

When Barney goes to college by bus, the probability that he will be late is 0.2 When Barney walks to college, the probability that he will be late is 0.1

Complete the probability tree diagram.

Barney will go to college on 200 days next year.

Work out an estimate for the number of days Barney will be late for college next year.

There are two bags of counters, bag X and bag Y.

There are 20 counters in bag X. 11 of the counters are blue and the rest are red.

There are 16 counters in bag Y. 9 of the counters are blue and the rest are red.

Arkady takes at random a counter from bag X and takes at random a counter from bag Y.

Complete the probability tree diagram.

bag X bag Y

Work out the probability that the two counters are both red.

Work out the probability that the two counters are both red or are both blue.

Jack plays a game with two fair spinners, A and B.

Spinner A can land on the number 2 or 3 or 5 or 7 Spinner B can land on the number 2 or 3 or 4 or 5 or 6

Jack spins both spinners. He wins the game if one spinner lands on an odd number and the other spinner lands on an even number.

Jack plays the game twice. Work out the probability that Jack wins the game both times.

Felix has 10 cards. There are 5 red cards, 4 blue cards and 1 green card.

Felix takes at random one of the cards. He does not replace the card. Felix then takes at random a second card.

Complete the probability tree diagram.

Work out the probability that Felix takes at least one blue card and no green card.