Combined & Conditional Probability

The probability that Sanay is late for school tomorrow is 
The probability that Jaden is late for school tomorrow is 

Alfie says that the probability that Sanay and Jaden will both be late for school tomorrow is  because

What assumption has Alfie made?

Shabeen has a biased coin.
 The probability that the coin will land on heads is 0.6

 Shabeen is going to throw the coin 3 times.

 She says the probability that the coin will land on tails 3 times is less than 0.1

 Is Shabeen correct?
 You must show all your working.

Here are seven tiles.

Jim takes at random a tile.
He does not replace the tile.

Jim then takes at random a second tile.

Calculate the probability that both the tiles Jim takes have the number 1 on them.

Calculate the probability that the number on the second tile Jim takes is greater than the number on the first tile he takes.

There are 9 counters in a bag.

7 of the counters are green.
2 of the counters are blue.

Ria takes at random two counters from the bag.

Work out the probability that Ria takes one counter of each colour.
You must show your working.

Fiza has 10 coins in a bag.
There are three £1 coins and seven 50 pence coins.

Fiza takes at random, 3 coins from the bag.

Work out the probability that she takes exactly £2.50

There are 7 blue counters, 3 green counters and 1 red counter in a bag.
There are no other counters in the bag.

Hubert takes at random 2 counters from the bag.

Work out the probability that both counters are blue.

Work out the probability that the 2 counters are different colours.

A box contains 15 counters.

There are 4 red counters, 5 green counters and the rest are yellow counters.

Niklas takes at random a counter from the box and writes down the colour of his counter.
He then puts the counter back into the box.

Sasha then takes at random a counter from the box and writes down the colour of her counter.

Work out the probability that the counters taken by Niklas and Sasha both have the same colour.

Barney has a biased coin.

When the coin is thrown once, the probability that the coin will land heads is 

Barney throws the coin  times.

Work out the probability that the coin will land heads exactly  times.

Work out the probability that the coin will land heads at least once.

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.

Sophie takes an examination.
If she fails the examination, she will resit.

The probability that Sophie passes the examination on her first attempt is 0.7
If she fails the examination on any attempt, the probability she passes on the next attempt is 0.9

Work out the probability that Sophie takes at most 2 attempts to pass the examination.

20 people were asked which device they used more often, laptop or phone.

The table shows the results.

One male and one female are chosen at random.

Work out the probability that exactly one of them said laptop.

Two males are chosen at random.

Work out the probability that they both said phone.

A bag contains 20 discs.

10 are red, 7 are blue and 3 are green.

Marnie takes a disc at random before putting it back in the bag.
Nick then takes a disc at random before putting it back in the bag.
Olly then takes a disc at random.

Work out the probability that they all take a red disc.

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.

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.

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.

Dice 2

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.

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 .

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)

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 = .........................................................