Possibility spaces and combined events

There are nine balls labelled 1 to 9 in a box. Lee will take at random two balls from the box. Lee says, “The probability that the sum of the numbers on the two balls will be an even number is greater than the probability that the product of the numbers will be an even number.” Is Lee correct? You must show how you get your answer.

There are 20 buttons in a box. There are 6 blue buttons 9 green buttons 5 red buttons. Jai puts 12 more buttons in the box. These buttons are either blue buttons or red buttons. Jai is going to pick at random one button from the box. The probability that this button will be blue is How many more red buttons did Jai put in the box? You must show all your working.

Freya and Ria each do an exam. The probability that Freya will pass the exam is 0.8 The probability that Ria will pass the exam is 0.85

Complete the probability tree diagram.

Work out the probability that only one of Freya or Ria will pass the exam.

There are only 7 blue pens and 3 red pens in a box. Raja takes at random one of the pens. He notes the colour of the pen and puts the pen back into the box. Raja does this two more times. Show that the probability that Raja takes at least two blue pens is

500 people were asked what type of film they liked best. 280 of the 500 people were adults. 100 of the adults said they liked action films best. 80 of the children said they liked thriller films best. 150 of the people said they liked action films best. 200 of the people said they liked comedy films best.

Complete the two-way table.

One of these 500 people is chosen at random.

Find the probability that this person is an adult who said they liked action films best.

In the s emi-finals of a chess tournament, player A will play player B and player C will play player D. The two winners will then play each other in the final. The probability that player A will win against player B is 0.6 The probability that player A will win against player C is 0.5 The probability that player A will win against player D is 0.3 The probability that player C will win against player D is 0.2 Work out the probability that player A will win the chess tournament.

Aisha has two boxes of sweets, box A and box B. In box A, there are only 10 red sweets and 30 green sweets. In box B, there are only 7 red sweets and 18 green sweets. Aisha is going to take at random a sweet from box A and a sweet from box B. Which box gives Aisha the greater probability of taking a red sweet, box A or box B? You must show how you get your answer.

22There are only red counters and yellow counters in a box. of the counters are red. Sophie takes at random two counters from the box. The probability that the two counters are the same colour is Work out the number of yellow counters in the box. You must show all your working.

Dan is playing cards. The probability that he will win a game of cards is 0.3 Dan plays two games of cards.

Complete the probability tree diagram.

Work out the probability that Dan does not win either game.

Dan is playing cards. The probability that he will win a game of cards is 0.3 Dan plays two games of cards.

Complete the probability tree diagram.

Work out the probability that Dan does not win either game.

There are only £10 notes and £20 notes in a wallet. Ali takes at random a note from the wallet.

Write down the probability that Ali takes a note with a value of more than £5

There are only 1p coins and 2p coins in a bag. The total value of the coins in the bag is 40p The total value of the 1p coins is the same as the total value of the 2p coins. Simon takes at random a coin from the bag.

Find the probability that Simon takes a 1p coin.

Here is a Venn diagram.

Write down the numbers that are in set P′

A number is chosen at random from the universal set,E

Find the probability that this number is in the set P Q

Here is a Venn diagram.

Write down the numbers that are in set P′

A number is chosen at random from the universal set,E

Find the probability that this number is in the set P Q

There are only n orange sweets and 1 white sweet in a bag. Saira takes at random a sweet from the bag and eats the sweet. She then takes at random another sweet from the bag and eats this sweet. Show that the probability that Saira eats two orange sweets is

There are only 4 red counters, 3 yellow counters and 1 green counter in a bag. Tony takes at random three counters from the bag. Work out the probability that there are now more yellow counters than red counters in the bag. You must show all your working.

Tim has two biased coins, coin A and coin B. He is going to throw both coins. The probability that coin A will land on heads is 0.6 The probability that coin B will land on heads is 0.55

Complete the probability tree diagram.

Tim throws coin A once and he throws coin B once.

Work out the probability that both coins land on heads.

Tim has two biased coins, coin A and coin B. He is going to throw both coins. The probability that coin A will land on heads is 0.6 The probability that coin B will land on heads is 0.55

Complete the probability tree diagram.

Tim throws coin A once and he throws coin B once.

Work out the probability that both coins land on heads.

Selina has a bag of 22 counters. 5 of the counters are blue. 9 of the counters are red. 8 of the counters are pink. Selina takes at random a counter from the bag. Write down the probability that Selina

takes a red counter,

does not take a pink counter,

takes a white counter.

Spinner A and spinner B are each spun once. The probability that spinner A lands on red is The probability that both spinner A and spinner B land on red is Work out the probability that one spinner lands on red and the other spinner does not land on red.

Shakir has to complete two tests. He can either pass or fail each test. The probability that he will pass the first test is 0.87 If he passes the first test the probability he will pass the second test is 0.94 If he fails the first test the probability he will pass the second test is 0.73

Complete the probability tree diagram for this information.

Work out the probability that Shakir passes at least one of the tests.

Zia has to set a 4‑digit security passcode on her phone. Each digit of the passcode is a number from 1 to 9 She can use each number more than once. Zia tells her friend Amber that the first digit is a cube number the second digit is a prime number the third digit is greater than 6 the fourth digit is an odd number. The diagram shows one possible 4‑digit passcode.

Amber is going to have one attempt at guessing Zia’s passcode. Work out the probability that Amber guesses Zia’s passcode on the first attempt.

E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {odd numbers} B = {square numbers}

Complete the Venn diagram for this information.

A number is chosen at random from the universal setE

Find the probability that this number is in the set B′

The probability tree diagram shows the probabilities that Shayla will work at home or will work at the office on two days next week.

Work out the probability that Shayla will work at home on Monday and work at the office on Friday.

E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {odd numbers} B = {square numbers}

Complete the Venn diagram for this information.

A number is chosen at random from the universal setE

Find the probability that this number is in the set B′

Martha plays a game twice. The probability tree diagram shows the probabilities that Martha will win or lose each game.

Find the probability that Martha will lose at least one game.

Here is a fair ordinary dice and a fair 8-sided spinner.

Charlie throws the dice once and spins the spinner once. Is Charlie more likely to get a number less than 3 on the dice or a number greater than 5 on the spinner? You must show all your working.

A biased dice is thrown 60 times. The table shows information about the number that the dice lands on each time.

Gethin throws the dice twice.

Work out an estimate for the probability that the dice will land on 6 both times.

Sally is going to throw the same dice n times and record the number it lands on each time. She will use her results to work out a more reliable estimate for the probability in part (a).

What can you say about the value of n?

There are only blue pens and red pens in a box. The number of blue pens is four times the number of red pens. Rita takes at random one pen from the box. She records the colour of the pen and then replaces it in the box. Rita does this n times, where n 2 Write down an expression, in terms of n, for the probability that Rita gets a blue pen at least once and a red pen at least once.

There is a total of y counters in a box. There are x pink counters and 5 blue counters in the box. The rest of the counters are green. x:y = 1: 3 Freda takes at random two counters from the box. Find, in terms of x, an expression for the probability that Freda takes two counters of the same colour.

Here is a list of 8 letters. B C A A A A B A

Write down the mode.

One of the 8 letters is going to be picked at random.

On the probability scale, mark with a cross () the probability that this letter will be B.

Find the probability that this letter will be C.

Lina spins a biased 5-sided spinner 40 times.

Here are her results.

Lina is now going to spin the spinner another two times.

Work out an estimate for the probability that she gets a score of 5 both times.

Derek is going to spin the spinner a large number of times.

Work out an estimate for the percentage of times Derek can expect to get a score of 1

Alfie has 11 cards. He has 3 blue cards 7 green cards and 1 white card. Alfie takes at random 2 of these cards. Work out the probability that he takes cards of different colours.

One weekend the Keddie family is going to do a sports quiz and a music quiz. The probability that the family will win the sports quiz is 0.3 The probability that the family will win the music quiz is 0.35

Complete the probability tree diagram. Sports quiz Music quiz

Work out the probability that the Keddie family will win both the sports quiz and the music quiz.

One weekend the Keddie family is going to do a sports quiz and a music quiz. The probability that the family will win the sports quiz is 0.3 The probability that the family will win the music quiz is 0.35

Complete the probability tree diagram. Sports quiz Music quiz

Work out the probability that the Keddie family will win both the sports quiz and the music quiz.

There are 15 sweets in a jar. 4 of the sweets are red. Jill takes at random a sweet from the jar.

Write down the probability that the sweet is red.

There are only green counters and blue counters in a bag. A counter is taken at random from the bag. The probability that the counter is green is 0.3

Find the probability that the counter is blue.

A first aid test has two parts, a theory test and a practical test. The probability of passing the theory test is 0.75 The probability of passing only one of the two parts is 0.36 The two events are independent. Work out the probability of passing the practical test.

Lorena gets a train at the same time each morning to go to work. She gets a train at the same time each evening to come home. The probability tree diagram shows the probabilities of each train arriving late.

Complete the probability tree diagram.

For a day that Lorena goes to work,

work out the probability that the train to work and the train home will both arrive late.

Here is a list of 8 numbers. 2 2 3 5 6 6 8 9 Kim picks at random one of these numbers.

On the probability scale below, mark with a cross () the probability that Kim picks a number 7

On the probability scale below, mark with a cross () the probability that Kim picks a number greater than 5

Find the probability that Kim picks an even number.

Ray has nine cards numbered 1 to 9

Ray takes at random three of these cards. He works out the sum of the numbers on the three cards and records the result. Work out the probability that the result is an even number.

There are only 3 red counters and 5 yellow counters in a bag. Jude takes at random 3 counters from the bag. Work out the probability that he takes exactly one red counter.

The pie chart gives information about the colour of each car in a car park.

There are 135 black cars in the car park.

Work out the number of white cars in the car park.

There are 50 grey cars in the car park. A car in the car park is picked at random.

Find the probability that this car is grey.

In a village, if it rains on one day, the probability that it will rain on the next day is 0.8 if it does not rain on one day, the probability that it will rain on the next day is 0.6 A weather forecaster says, “There is a 70% chance that it will rain in the village on Monday.” Work out an estimate for the probability that it will rain in the village on Wednesday. You must show all your working.

Pat throws a fair coin n times. Find an expression, in terms of n, for the probability that Pat gets at least 1 head and at least 1 tail.

Sally plays two games against Martin. In each game, Sally could win, draw or lose. In each game they play, the probability that Sally will win against Martin is 0.3 the probability that Sally will draw against Martin is 0.1 Work out the probability that Sally will win exactly one of the two games against Martin.

E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {even numbers} B = {factors of 10}

Complete the Venn diagram for this information.

A number is chosen at random from the universal set, E

Find the probability that this number is in the set A B

E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {even numbers} B = {factors of 10}

Complete the Venn diagram for this information.

A number is chosen at random from the universal set, E

Find the probability that this number is in the set A B

There are only red sweets and yellow sweets in a bag. There are n red sweets in the bag. There are 8 yellow sweets in the bag. Sajid is going to take at random a sweet from the bag and eat it. He says that the probability that the sweet will be red is

Show why the probability cannot be

After Sajid has taken the first sweet from the bag and eaten it, he is going to take at random a second sweet from the bag. Given that the probability that both the sweets he takes will be red is

work out the number of red sweets in the bag. You must show all your working.

There are only 5 blue cards, 2 green cards and 4 red cards in a pack. Isabella is going to take at random one card from the pack.

Write down the probability that Isabella will take a blue card.

Ken is going to throw a biased dice once. The probability that the dice will land on six is 0.3

What is the probability that the dice will not land on six?

Amanda has two fair 3-sided spinners.

Amanda spins each spinner once.

Complete the probability tree diagram.

Work out the probability that Spinner A lands on 2 and Spinner B does not land on 2

Amanda has two fair 3-sided spinners.

Amanda spins each spinner once.

Complete the probability tree diagram.

Work out the probability that Spinner A lands on 2 and Spinner B does not land on 2

Marek has 9 cards. There is a number on each card.

Marek takes at random two of the cards. He works out the product of the numbers on the two cards.

Work out the probability that the product is an even number.

The probability tree diagram shows the probabilities that Bismah will be late for work on two days next week.

Calculate the probability that Bismah will be late on exactly one of the two days.

Work out the probability of getting 4 tails when the coin is thrown 4 times.

Mary travels to work by train every day. The probability that her train will be late on any day is 0.15

Complete the probability tree diagram for Thursday and Friday.

Work out the probability that her train will be late on at least one of these two days.

E {1, 2, 3, 4, 5, 6, 7, 8, 9} A {1, 5, 6, 8, 9} B {2, 6, 9}

Complete the Venn diagram to represent this information.

A number is chosen at random from the universal set E.

Find the probability that the number is in the set A B

E {1, 2, 3, 4, 5, 6, 7, 8, 9} A {1, 5, 6, 8, 9} B {2, 6, 9}

Complete the Venn diagram to represent this information.

A number is chosen at random from the universal set E.

Find the probability that the number is in the set A B

Amina has two bags.

In the first bag there are 3 red balls and 7 green balls. In the second bag there are 5 red balls and 4 green balls.

Amina takes at random a ball from the first bag. She then takes at random a ball from the second bag.

Complete the probability tree diagram.

Work out the probability that Amina takes two red balls.

There are only green pens and blue pens in a box.

There are three more blue pens than green pens in the box. There are more than 12 pens in the box.

Simon is going to take at random two pens from the box. The probability that Simon will take two pens of the same colour is Work out the number of green pens in the box.

There are 49 counters in a bag.

20 of the counters are red. The rest of the counters are blue.

One of the counters is taken at random.

Find the probability that the counter is blue.

Victoria throws an ordinary fair 6-sided dice once.

She says,

“The probability of getting a 3 is half the probability of getting a 6”

Is Victoria correct? You must explain your answer.

Andy throws the dice twice. He says, “The probability of getting a 6 on both throws is ”

Is Andy correct? You must explain your answer.

Indre throws the dice once. She also throws a coin to get Heads or Tails.

List all the possible outcomes she can get.

E = {even numbers between 1 and 25} A = {2, 8, 10, 14} B = {6, 8, 20} C = {8, 18, 20, 22}

Complete the Venn diagram for this information.

A number is chosen at random from E.

Find the probability that the number is a member of A B.

E = {even numbers between 1 and 25} A = {2, 8, 10, 14} B = {6, 8, 20} C = {8, 18, 20, 22}

Complete the Venn diagram for this information.

A number is chosen at random from E.

Find the probability that the number is a member of A B.

There are only red counters and blue counters in a bag.

Joe takes at random a counter from the bag. The probability that the counter is red is 0.65 Joe puts the counter back into the bag.

Mary takes at random a counter from the bag. She puts the counter back into the bag.

What is the probability that Joe and Mary take counters of different colours?

There are 78 red counters in the bag.

How many blue counters are there in the bag?

An ordinary fair dice is thrown once.

On the probability scale below, mark with a cross () the probability that the dice lands on an odd number.

Write down the probability that the dice lands on a number greater than 4

A scout group has a raffle to raise money for charity. There is 1 prize to be won in the raffle.

Laura buys 12 raffle tickets. A total of 350 raffle tickets are sold.

Find the probability that Laura does not win the prize.

A darts team is going to play a match on Saturday and on Sunday. The probability that the team will win on Saturday is 0.45

If they win on Saturday, the probability that they will win on Sunday is 0.67 If they do not win on Saturday, the probability that they will win on Sunday is 0.35

Complete the probability tree diagram.

Find the probability that the team will win exactly one of the two matches.

When a biased 6-sided dice is thrown once, the probability that it will land on 4 is 0.65 The biased dice is thrown twice.

Amir draws this probability tree diagram. The diagram is not correct.

Write down two things that are wrong with the probability tree diagram.

When a biased 6-sided dice is thrown once, the probability that it will land on 4 is 0.65 The biased dice is thrown twice.

Amir draws this probability tree diagram. The diagram is not correct.

Write down two things that are wrong with the probability tree diagram.

50 people were asked if they speak French or German or Spanish.

Of these people, 31 speak French 2 speak French, German and Spanish 4 speak French and Spanish but not German 7 speak German and Spanish 8 do not speak any of the languages all 10 people who speak German speak at least one other language

Two of the 50 people are chosen at random.

Work out the probability that they both only speak Spanish.

There are 3 cards in Box A and 3 cards in Box B. There is a number on each card.

Ryan takes at random a card from Box A and a card from Box B. He adds together the numbers on the two cards to get a total score.

Work out the probability that the total score is an odd number.

There are only blue cubes, yellow cubes and green cubes in a bag.

There are

twice as many blue cubes as yellow cubes and four times as many green cubes as blue cubes.

Hannah takes at random a cube from the bag.

Work out the probability that Hannah takes a yellow cube.

There are 12 counters in a bag. There is an equal number of red counters, blue counters and yellow counters in the bag. There are no other counters in the bag.

3 counters are taken at random from the bag.

Work out the probability of taking 3 red counters.

The 3 counters are put back into the bag.

Some more counters are now put into the bag. There is still an equal number of red counters, blue counters and yellow counters in the bag. There are no counters of any other colour in the bag.

3 counters are taken at random from the bag.

Is it now less likely or equally likely or more likely that the 3 counters will be red? You must show how you get your answer.

When a drawing pin is dropped it can land point down or point up.

Lucy, Mel and Tom each dropped the drawing pin a number of times.

The table shows the number of times the drawing pin landed point down and the number of times the drawing pin landed point up for each person.

Rachael is going to drop the drawing pin once.

Whose results will give the best estimate for the probability that the drawing pin will land point up? Give a reason for your answer.

Stuart is going to drop the drawing pin twice.

Use all the results in the table to work out an estimate for the probability that the drawing pin will land point up the first time and point down the second time.

When a drawing pin is dropped it can land point down or point up.

Lucy, Mel and Tom each dropped the drawing pin a number of times.

The table shows the number of times the drawing pin landed point down and the number of times the drawing pin landed point up for each person.

Rachael is going to drop the drawing pin once.

Whose results will give the best estimate for the probability that the drawing pin will land point up? Give a reason for your answer.

Stuart is going to drop the drawing pin twice.

Use all the results in the table to work out an estimate for the probability that the drawing pin will land point up the first time and point down the second time.

Alan has two spinners, spinner A and spinner B. Each spinner can land on only red or white.

The probability that spinner A will land on red is 0.25 The probability that spinner B will land on red is 0.6

The probability tree diagram shows this information.

Alan spins spinner A once and he spins spinner B once. He does this a number of times.

The number of times both spinners land on red is 24

Work out an estimate for the number of times both spinners land on white.

Sammy spins a fair 4-sided spinner.

On the probability scale, mark with a cross (×) the probability that the spinner will land on B.

On the probability scale, mark with a cross (×) the probability that the spinner will land on F.

There are only 7 blue pens, 4 green pens and 6 red pens in a box.

One pen is taken at random from the box.

Write down the probability that this pen is blue.

There are 30 students in Mr Lear’s class. 16 of the students are boys.

Two students from the class are chosen at random.

Mr Lear draws this probability tree diagram for this information.

Write down one thing that is wrong with the probabilities in the probability tree diagram.

Owen and Wasim play for the school football team.

The probability that Owen will score a goal in the next match is 0.4 The probability that Wasim will score a goal in the next match is 0.25

Mr Slater says,

“The probability that both boys will score a goal in the next match is 0.4 + 0.25”

Is Mr Slater right? Give a reason for your answer.

E = {odd numbers less than 30} A = {3, 9, 15, 21, 27} B = {5, 15, 25}

Complete the Venn diagram to represent this information.

A number is chosen at random from the universal set, E.

What is the probability that the number is in the set A B?

E = {odd numbers less than 30} A = {3, 9, 15, 21, 27} B = {5, 15, 25}

Complete the Venn diagram to represent this information.

A number is chosen at random from the universal set, E.

What is the probability that the number is in the set A B?