Combined & Conditional Probability (OCR GCSE Maths)

Four friends each throw a biased coin a number of times.
The table shows the number of heads and the number of tails each friend got.

The coin is to be thrown one more time.

Which of the four friends' results will give the best estimate for the probability that the coin will land heads?
Justify your answer.

Paul says,

"With this coin you are twice as likely to get heads as to get tails."

Is Paul correct?
Justify your answer.

The coin is to be thrown twice.

Use all the results in the table to work out an estimate for the probability that the coin will land heads both times.

In a supermarket, the probability that John buys fruit is 0.7

In the same supermarket, the probability that John independently buys vegetables is 0.4

Work out the probability that John buys fruit or buys vegetables or buys both.

Thelma spins a biased coin twice.
The probability that it will come down heads both times is 0.09

Calculate the probability that it will come down tails both times.

Here are 9 cards.
Each card has a shape on it.

In a game the cards are turned over so that the shapes are hidden.
The cards are then mixed up.

Katie turns over at random two of the cards.

Work out the probability that these two cards have different shapes on them.
You must show all your working.

Carolyn has 20 biscuits in a tin.

  She has

12 plain biscuits
5 chocolate biscuits
3 ginger biscuits

  Carolyn takes at random two biscuits from the tin.

  Work out the probability that the two biscuits were not the same type.

There are three different types of sandwiches on a shelf.

  There are

  Erin takes at random 2 of these sandwiches.

  Work out the probability that she takes 2 different types of sandwiches.

Nomusa has 30 sweets.

  She has

18 fruit sweets
7 aniseed sweets
5 mint sweets

  Nomusa is going to take at random two sweets.

  Work out the probability that the two sweets will not be the same type of sweet.
  You must show all your working.

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.

There are 10 pens in a box.

There are  red pens in the box.
All the other pens are blue.

Jack takes at random two pens from the box.

Find an expression, in terms of , for the probability that Jack takes one pen of each colour.
Give your answer in its simplest form.

Students are asked to choose one subject from Option A and one subject from Option B.

If a student chooses their subjects at random, what is the probability that both subjects have the same first letter?

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.

Hector has a bag that contains 12 counters.
There are 7 green counters and 5 red counters in the bag.

Hector takes at random a counter from the bag.
He looks at the counter and puts the counter back into the bag.

Hector then takes at random a second counter from the bag.
He looks at the counter and puts the counter back into the bag.

Complete the probability tree diagram.

Work out the probability that both counters are red.

Meghan has a jar containing 15 counters.
There are only blue counters, green counters and red counters in the jar.

Hector is going to take at random one of the counters from his bag of 12 counters.
He will look at the counter and put the counter back into the bag.

Hector is then going to take at random a second counter from his bag.
He will look at the counter and put the counter back into the bag.

Meghan is then going to take at random one of the counters from her jar of counters.
She will look at the counter and put the counter back into the jar.

The probability that the 3 counters each have a different colour is 

Work out how many blue counters there are in the jar.

Jack plays a game with two fair spinners, and

Spinner can land on the number 2 or 3 or 5 or 7
Spinner 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.

A bag contains 8 balls.

3 are red and 5 are blue.

2 balls are taken from the bag at random without replacement.

Write down the probability that there is at least 1 red ball still in the bag.

Work out the probability that there are at least 2 red balls still in the bag.

David has designed a game.
He uses a fair 6-sided dice and a fair 5-sided spinner.
The dice is numbered 1 to 6
The spinner is numbered 1 to 5

Each player rolls the dice once and spins the spinner once.
A player can win £5 or win £2

David expects 30 people will play his game.
Each person will pay David £1 to play the game.

Work out how much profit David can expect to make.

Give a reason why David's actual profit may be different to the profit he expects to make.

Paul has 8 cards.
  There is a number on each card.

  Paul takes at random 3 of the cards.
  He adds together the 3 numbers on the cards to get a total

  Work out the probability that is an odd number.

John has an empty box.
He puts some red counters and some blue counters into the box.

The ratio of the number of red counters to the number of blue counters is 

Linda takes at random  counters from the box.

The probability that she takes  red counters is 

How many red counters did John put into the box?

There are  black socks and  white socks in a drawer.

Joshua takes at random two socks from the drawer.

The probability that Joshua takes one white sock and one black sock is 

Show that

Find the probability that Joshua takes two black socks.

There are  sweets in a bag.
   of the sweets are orange.
  The rest of the sweets are yellow.

  Hannah takes at random a sweet from the bag.
  She eats the sweet.

  Hannah then takes at random another sweet from the bag.
  She eats the sweet.

  The probability that Hannah eats two orange sweets is 
  Show that 

Solve to find the value of .

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 .

t = .........................................................

Magnus 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.

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.

In a bag, there are only

 blue beads
 white beads
and  orange beads.

Jean is going to take at random two beads from the bag.
The probability that Jean will take two beads of the same colour is 

Find the total number of beads in the bag.
Show clear algebraic working.

Pippa has a box containing pens.

There are only black pens and red pens in the box.
The number of black pens in the box is 3 more than the number of red pens.

Pippa is going to take at random 2 pens from the box.

The probability that she will take a black pen followed by a red pen is

Find the possible values of .
Show clear algebraic working.

A bowl contains  pieces of fruit.
Of these,  are oranges and the rest are apples.

Two pieces of fruit are going to be taken at random from the bowl.
The probability that the bowl will then contain  apples is 

Work out the value of 
Show your working clearly.

Liam is trying to remember a 3-digit code.

He knows the rule that

the first digit is a cube number
the second digit is a factor of 16
the third digit is an odd number.

Liam tries at random a code that matches the rule.

Work out the probability that this is the correct code.

These 20 discs are in a bag.

Two of the discs are taken at random from the bag.

Work out the probability that the first disc has a smaller number than the second disc.

A bag contains 30 discs.

10 are red and 20 are blue.

   One disc is taken out at random and replaced by two of the other colour.

   Another disc is then taken out at random and replaced by two of the other colour.

   Another disc is then taken out at random.

Work out the probability that all three discs taken out are red.

Bag X contains 9 blue balls and 18 red balls.
Bag Y contains 7 blue balls and 14 red balls.

Liz picks a ball at random from bag X.
She puts the ball into bag Y.
Mike now picks a ball at random from bag Y.

Show that

   P (Liz picks a blue ball) = P (Mike picks a blue ball)