Conditional probability

Casey plays two games of football. In each game, the probability that Casey scores a goal is 0.3.

Complete the tree diagram.

First game
Second game

Work out the probability that Casey scores in only one of the games.

A school has 90 students in Year 11.

• 50 of the students study French.

• 59 of the students study Spanish.

• 4 of the students do not study either of these languages.

One the students is chosen at random.

Find the probability that the student studies French given that they study Spanish. You may use the grid or the Venn diagram to help present your working if you wish.

There are 10 sweets in a bag. 4 of the sweets are red, 4 are green and 2 are yellow.

Charlie says,

The probability of picking the two yellow sweets at random from the bag can be found

Write down the assumption that Charlie has made.

Mia takes one of the 10 sweets from the bag at random and eats it. Ben then takes a sweet from the bag and eats it.

Find the probability that the bag now holds more red sweets than green sweets. You must show your working.

A packet of 6 pens contains 4 red pens and 2 blue pens.

A pen is taken at random from the packet and is not replaced. A second pen is then taken at random from the packet.

Complete the tree diagram.

First pen
Second pen

Find the probability of taking two pens of the same colour.

Kai spins a fair five-sided spinner. The sectors of the spinner are numbered 1, 2, 3, 4 and 5.

Kai also throws a fair four-sided dice. Two of the dice faces are red, one is blue, and one is yellow.

Complete this tree diagram.

Calculate the probability that Kai gets an even number on the spinner and a blue face on the dice.

In a survey, some students were asked whether they had travelled to school by bus (B) or by car (C) in the last week. The Venn diagram shows some of the results.

One of the students is chosen at random. The probability that, in the last week, this student had travelled to school by bus and by car

Find the value of n.

One of the students is chosen at random.

Find the probability that, in the last week, this student had travelled to school by car given that they had also travelled to school by bus.

A student attempts two tasks. The result of each task is either “Pass” or “No pass”.

The probability of the student passing the first task is 0.6. The probability of the student passing the second task is x.

Complete the tree diagram.

First task
Second task

Write down the mathematical assumption that has been made about the two tasks.

The probability of the student passing just one of these two tasks is 0.528.

Work out the value of x.

The box plot shows the distribution of the waiting time of cars at a road junction.

Write down the median waiting time.

What percentage of the waiting times were less than 30 seconds?

Given that a randomly chosen car waited for more than 30 seconds, write down the probability that the car waited for more than 90 seconds.

In a survey, 50 people are asked which sports they watch. The results are shown on the Venn diagram.

One person is chosen at random from those that watch athletics.

Find the probability that this person watches only one other sport.

Two of the 50 people are chosen at random.

Show that the probability that one of them watches only football and the other watches only

On a bus route there is a set of traffic lights and a level crossing. The probability that the bus stops at the traffic lights is 0.7. The probability that the bus stops at the level crossing is 0.2.

Use the information to complete the tree diagram.

Find the probability that the bus stops at either the traffic lights or at the level crossing but not both.

In a computer game the player can either win or lose. A student thinks the ratio of the probability of winning to the probability of losing is 2: 3.

The student plays two games.

Use the information to complete the tree diagram.

Find the probability that the student wins at least one of the two games.

The student now thinks the ratio of the probability of winning to the probability of losing has changed to 2: 5.

Explain the effect this change will have on your answer to part (b).

Riley has a set of cards. Each card has a triangle or circle drawn on it and is coloured red or blue.

The table gives some information about the number of each type of card.

Riley chooses three of these cards at random without replacement. All three of these cards have a circle drawn on them.

Riley says The probability that two of these three cards are blue and one is red is less than Is Riley correct? You must show your working.

A cafe owner recorded information about customer orders for coffee. They recorded whether the customer asked for milk (M) and whether the customer asked for sugar (S).

The results are shown in this Venn diagram.

One of the customers is chosen at random.

Find the probability that the customer asked for sugar.

One of the customers is chosen at random.

Find the probability that the customer asked for sugar given that they asked for milk.

The probability that Sam works from home on Monday is 0.4. The probability that Sam works from home on Friday is 0.2.

Complete the tree diagram.

Work out the probability that Sam works from home on Monday but does not work from home on Friday.

100 people were asked whether they had visited France (F) or Spain (S).

• 55 had visited France

• 60 had visited Spain

• 4 had not visited either country.

Complete the Venn diagram.

One of these 100 people is chosen at random.

Write down the probability that this person had visited exactly one of the countries.

Write down the probability that this person had visited France given that they had also visited Spain.

If it rains on a given day the probability that it will rain the next day is 0.65. If it does not rain on a given day the probability that it will rain the next day is 0.3. It rained on Monday.

Complete the tree diagram.

Find the probability that it rains on Wednesday.

A box contains 25 discs. The discs are either blue or yellow in the ratio 4: 1. Two discs are chosen at random from the box without replacement.

Find the probability that the two discs are different colours. You must show your working.

Alex has a bag containing 3 blue beads and 5 green beads. There are no other beads in the bag.

Alex takes a bead at random from the bag, puts it back, and then takes another bead.

Alex says

The probability that the two beads are the same colour is less than 50%.

Is Alex correct? Show how you decide. You may use this tree diagram if you wish.

Sam and Taylor are playing a game against a computer. They can win, draw or lose the game.

Sam says

I think the probability of us winning the game is 0.3.

Taylor says

I think the probability of us losing the game is 0.75.

Explain why Sam and Taylor cannot both be correct.

Sam is correct. The probability of them winning the game is 0.3. Taylor is not correct. The probability of them losing the game is actually 0.55.

Complete this partly drawn tree diagram to show all the possible outcomes of playing the game twice.

Find the probability of them winning the first game and losing the second game.

A salesroom sells various types of car. Some cars are electric (E), some are petrol (P), some are both and some are neither.

The Venn diagram below shows the salesroom’s stock of cars.

A petrol car is picked at random.

Find the probability that the car is also electric.

The time, t seconds, taken by each of 60 students to complete a puzzle is recorded.

The table shows information about these times.

Two students are picked at random. Reece works out the probability that they both took longer than 50 seconds to complete the puzzle. Reece’s working is shown below.

Explain the error in their method and write the correct calculation that Reece needs to do.

You do not need to work out the answer to the calculation.

The correct calculation is........

Two students are picked at random from those who took 50 seconds or less.

Find the probability that one of them took 30 seconds or less and the other took more than 40 seconds. You must show your working.

Sam and Taylor are playing a game against a computer. They can win, draw or lose the game.

Sam says

I think the probability of us winning the game is 0.3.

Taylor says

I think the probability of us losing the game is 0.75.

Explain why Sam and Taylor cannot both be correct.

Sam is correct. The probability of them winning the game is 0.3. Taylor is not correct. The probability of them losing the game is actually 0.55.

Complete this partly drawn tree diagram to show all the possible outcomes of playing the game twice.

Find the probability of them winning the first game and losing the second game.

Reece travels to school by either walking, cycling or using a bus. The probability that Reece walks is always 0.4. The probability that Reece cycles is always 0.55.

Complete the tree diagram for Monday and Tuesday.

Show that the probability that Reece travels to school by the same method on Monday and

Tuesday is 0.465.

There are 15 sweets in a bag. 10 of the sweets are toffee and 5 are mint. Reece takes two of the sweets at random.

Work out the probability that Reece takes one of each type of sweet.

In a survey about music, some students were asked whether they like pop (P) and whether they like rap (R).

The Venn diagram shows some of the results. x students liked both types of music.

The ratio of the number of students who liked pop to the number who liked rap was 5: 2.

Work out the total number of students in the survey.

One of the students is selected at random.

Find the probability that this student does not like rap given that they like pop.

A student has a spinner with sectors numbered 1, 2, 3 and 4.

The table shows the probability of each score.

The student spins the spinner twice.

Calculate the probability that the student gets the same score on each spin.

Over a long period of time, it is found that the probability of a train from Bewford to London being late is 0.2.

One morning there are two trains from Bewford to London.

Use the information to complete the tree diagram.

Work out the probability that both trains are not late.

Give a reason why the probabilities used in the tree diagram for the second train may not be reliable.

Morgan takes a train from London to Bewford and then another train to Agon. The tree diagram shows the probabilities of Morgan’s trains being late or not late.

Morgan will not catch the train to Agon if the train to Bewford is late and the train to Agon is not late.

Work out the probability that Morgan will catch the train to Agon.

Over a long period of time, it is found that the probability of a train from Bewford to London being late is 0.2.

One morning there are two trains from Bewford to London.

Use the information to complete the tree diagram.

Work out the probability that both trains are not late.

Give a reason why the probabilities used in the tree diagram for the second train may not be reliable.

Morgan takes a train from London to Bewford and then another train to Agon. The tree diagram shows the probabilities of Morgan’s trains being late or not late.

Morgan will not catch the train to Agon if the train to Bewford is late and the train to Agon is not late.

Work out the probability that Morgan will catch the train to Agon.

In a group of 60 students, 40 own a smartphone, 27 own a tablet and 8 own neither.

A student is chosen at random from those that own a tablet.

Find the probability that they also own a smartphone. You must show your working.

In a class of 30 students

• 17 study French (F)

• 20 study German (G)

• 2 do not study either subject.

Complete the Venn diagram.

Two of the 30 students are chosen at random.

Calculate the probability that one of these two students studies French but not German and the other studies German but not French. You must show your working.

The probability that Adam cycles to school or walks to school depends on the weather.

• On any day, the probability that the weather is wet is 0.4.

• When the weather is wet the probability that he cycles to school is 0.3.

• When the weather is dry the probability that he cycles to school is 0.8.

The information is shown on this tree diagram.

The weather
Getting to school

Work out the probability that

it is dry and Adam walks to school,

Adam cycles to school.

Li has t toy bricks. She only has red bricks and blue bricks.

Li picks two bricks, one after the other. If the first brick she picks is red, the probability that the second brick is red is If the first brick she picks is blue, the probability that the second brick is red is Calculate the value of t.

A bus company has a large number of buses. 25% of the buses are more than 10 years old.

If a bus is more than 10 years old, the probability that it will start first time is 0.3. If a bus is less than 10 years old, the probability that it will start first time is 0.65.

Amir is asked to drive one of the company’s buses, chosen at random.

Calculate the probability that the bus starts first time.

Ryan shoots an arrow at a target. He then kicks a ball at a goal. The probability that Ryan hits the target is The probability that Ryan scores a goal is

Complete the tree diagram.

........

Find the probability that Ryan

misses the target and does not score a goal,

either hits the target or scores a goal or both.

Jacob, Amelie and Reuben each roll a fair six-sided dice. What is the probability that all three roll a number less than 3?

Give your answer as a fraction in its simplest form.

50 people attended an outdoor activity day.

• 40 took part in walking.

• 18 took part in sailing.

• 3 did neither activity.

One of the people who walked is chosen at random.

Find the probability that this person also sailed.

Jacob, Amelie and Reuben each roll a fair six-sided dice. What is the probability that all three roll a number less than 3?

Give your answer as a fraction in its simplest form.

Dani has a pack of 45 cards. Each card is either red or black.

One-third of the cards in the pack are red.

She picks two cards from the pack, without replacement.

Calculate the probability that Dani picks two black cards.

A bag contains 4 red counters and 3 blue counters only. Jack picks a counter at random and then replaces it. Jack then picks a second counter at random.

Complete the tree diagram.

Work out the probability that Jack picks two red counters.

A bag contains 4 red counters and 3 blue counters only. Jack picks a counter at random and then replaces it. Jack then picks a second counter at random.

Complete the tree diagram.

Work out the probability that Jack picks two red counters.

21 people travelled to a meeting.

• 12 used a train.

• 6 used a car.

• 7 did not use a train or a car.

• Some used a train and a car.

Two people are chosen at random from those who used a train.

Find the probability that both these people also used a car.

Antonio rolls two fair six-sided dice and calculates the difference between the scores. For example, if the two scores are 2 and 5 or 5 and 2 then the difference is 3.

Complete the sample space diagram to show the possible outcomes from Antonio’s dice.

Antonio rolls the two dice three times.

Calculate the probability that he gets a difference of 1 on all three rolls. Give your answer as a fraction in its lowest terms.

Romelu picks a shirt and shorts. The probability he picks a red shirt is 0.4. The probability he picks white shorts is 0.7.

Complete the tree diagram.

Calculate the probability that Romelu picks a red shirt but does not pick white shorts.

The probability that any postcard posted in Portugal on Monday is delivered to the UK within a week is 0.62. The probability that any postcard posted in Portugal on Friday is delivered to the UK within a week is 0.41.

Anna is on holiday in Portugal. She posts 15 postcards to the UK on Monday.

How many of her postcards can she expect to be delivered within a week?

Sergio is in Portugal. He posts one postcard to the UK on Monday. He posts another postcard to the UK on Friday.

Complete the probability tree to show the possible outcomes for the postcards.

Monday’s Friday’s postcard postcard

Calculate the probability that only one of Sergio’s postcards is delivered within a week.

Finn has two bags of counters. He takes a counter at random from each bag.

The probability that he takes a red counter from the first bag is 0.3. The probability that he takes a red counter from the second bag is 0.4.

What is the probability that he takes at least one red counter?

The Venn diagram shows the number of students studying Mathematics (M) and the number of students studying Physics (P) in a college. 35 students do not study either subject.

The total number of students is 121.

Find the value of x.

One of the 121 students is selected at random.

Find the probability that this student studies Mathematics, given that they study Physics.

Noah starts to draw a tree diagram showing the outcomes of throwing a six when a fair dice is thrown twice.

Complete the tree diagram.

What is the probability of throwing two sixes?

Cara throws the same dice three times. Show that the probability that Cara does not throw a six until her third throw is

Adam has 10 sweets in a bag. 5 are cherry sweets, 4 are lemon sweets and 1 is an orange sweet.

Adam chooses a sweet at random from the bag and eats it. He then takes another sweet at random from the bag and eats it.

Adam says The probability that I choose two cherry sweets is He is incorrect. Explain his error.

Find the probability that the two sweets he chooses have different flavours.

Complete this tree diagram to show the probabilities for each card picked in the game.

What is the probability that Reuben wins the game?

Kirsty either travels by bus or walks when she visits the shops. The probability that she catches the bus to the shops is 0.3. The probability that she catches the bus from the shops is 0.8.

To the shops
From the shops

Complete the tree diagram.

Show that the probability that Kirsty walks at least one way is 0.76.

Rashid drives his car along a road passing through two sets of traffic lights. The tree diagram shows the probabilities of the lights being red when he reaches them.

Complete the tree diagram.

Write down the probability that the first set is not red.

Given that the first set is red, write down the probability that the second set is not red.

Work out the probability that both sets are not red.

Work out the probability that at least one set is not red.

In a group of 120 adults, 85 watch football, 78 play a sport and 20 do neither.

Find the probability that an adult chosen at random from those who watch football does not play a sport.