The RobertSmythSchoolTopic 1
Mathematics FacultyProbability
Probability and Expectation
1.Alan, Bob and Colin play a game of darts.
There is only one winner.
The probability that Alan wins the game is 0.3
The probability that Bob wins the game is 0.5
(a)Write down the probability that Alan does not win the game.
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(1)
(a)What is the probability that Alan or Bob wins the game?
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(1)
(b)Alan, Bob and Colin play 20 games of darts.
How many games would you expect Colin to win?
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(4)
(Total 6 marks)
2.A game is played with a coloured spinner.
The arrow is spun once.
The table shows some of the probabilities of the arrow landing on a colour.
Red / 0.4
Blue / 0.2
Green / 0.3
White
(a)Calculate the probability that the arrow lands on red or blue.
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(2)
(b)Calculate the probability that the arrow lands on white.
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(2)
(c)The arrow is spun 250 times.
Calculate the number of times you would expect the arrow to land on red or green.
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(3)
(Total 7 marks)
3.Red, blue, white and green tickets are sold in a raffle.
The table shows some of the probabilities of these tickets winning the first prize.
first prize
Red / 0.4
Blue / 0.2
White / 0.1
Green
(a)Calculate the probability of a green ticket winning the first prize.
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Answer ...... (2)
(b)There were 1000 tickets sold in this raffle.
Calculate how many red tickets and blue tickets were sold altogether.
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(2)
(Total 4 marks)
4.A school has 1260 students.
The probability that a student at the school has blue eyes is .
Calculate how many students at the school have blue eyes.
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(Total 2 marks)
5.In a survey about favourite methods of travel people could choose car, train, coach or aeroplane. The following probabilities were calculated from the results.
Method of travel / ProbabilityCar / 0.45
Train
Coach / 0.17
Aeroplane / 0.12
200 people took part in this survey.
How many chose train?
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(Total 4 marks)
6.A bag contains 200 coloured discs.
The discs are either red, blue or yellow.
There are 86 red discs in the bag.
The probability that a blue disc is chosen from the bag is 0.22
Calculate the number of yellow discs in the bag.
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(Total 4 marks)
7A scout group organises a game to raise money.
200 people each pay £2 to play the game.
The probability that a person wins is .
The winners each receive £5 and there are no other prizes.
Calculate how much profit the scout group makes from this game.
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Answer £ ......
(Total 4 marks)
Relative Probability
1.Here are three statements about probability.
Tick a box to show whether you agree or disagree with each statement.
Give a reason for each answer.
(a)Graham says, “The probability that it will rain tomorrow is ”.
Reason ......
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(1)
(b)Mandy says, “In my box of chocolates there are 13 soft centres and 15 hard centres so the probability of my choosing a soft centre is “.
Reason ......
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(1)
(c)Tom tosses a fair coin twice.
He gets a head both times.
He says, “The probability that I will get a head the next time I throw the coin is ”.
Reason ......
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(1)
(Total 3 marks)
2.A dice is suspected of bias.
Here are the results of 20 throws.
4 / 3 / 1 / 1 / 6 / 2 / 5 / 6 / 5 / 3
(a)Use these results to calculate the relative frequency of each score.
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Score / 1 / 2 / 3 / 4 / 5 / 6Relative frequency
(2)
(b)Use the relative frequency to calculate how many times you would expect to score 3 in 60 throws of this dice.
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(2)
(c)Compare your answer to part (b) with the number of times you would expect to score 3 in 60 throws of a fair dice.
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(1)
(Total 5 marks)
3.Twenty pupils each shuffle a pack of coloured cards and choose a card at random.
The colour of the card is recorded for each pupil.
(R = RedB = BlueG = GreenY = Yellow)
BG
Y
B / Y
R
R
B / Y
Y
B
G / G
B
B
R / R
B
Y
Y
(a)Use these results to calculate the relative frequency of each colour.
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Colour / Red / Blue / Green / YellowRelative frequency
(2)
(b)Use the results to calculate how many times you would expect a blue card if 100 pupils each choose a card at random.
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(2)
(Total 4 marks)
4.Ronnie is a snooker player.
He takes 20 practice shots at potting the black ball.
The table shows whether he pots the black ball () or misses (×) on each shot.
Result / × / × / / × / / × / / / × /
Shot number / 11 / 12 / 13 / 14 / 15 / 16 / 17 / 18 / 19 / 20
Result / × / / / / × / / / × / / ×
(a)Write down the relative frequency of Ronnie potting the black ball by using the resultsfrom
(i)his first five shots
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(1)
(ii)his first ten shots.
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(b)Ronnie states that, on average, he can pot the black ball on more than 50% of his shots.
Explain how the results from the table support his statement.
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In a match the first player to win 8 games is the winner.
The scatter diagram shows the number of games that Ronnie won in each match and the number of hours that he practised before each match.
(c)Tick the box which best describes the strength and type of correlation shown in the scatter diagram.
Strong WeakNoWeak Strong
negativenegativecorrelationpositivepositive
(1)
(d)Ronnie draws a line of best fit on the scatter diagram.
He uses it to estimate the number of games he would win if he practised for 4.5 hours before a match.
Explain why this is not sensible.
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(1)
(Total 5 marks)
5.(a)Matthew has a dice with 3 red faces, 2 blue faces and 1 green face.
He throws the dice 300 times.
The results are shown in the table.
153 / 98 / 49
(i)What is the relative frequency of throwing a red?
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(1)
(ii)Is the dice fair?
Explain your answer.
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(2)
(b)Emmie has a dice with 4 red faces and 2 blue faces.
She throws the dice 10 times and gets 2 reds.
Emmie says the dice is not fair.
Explain why Emmie could be wrong.
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(1)
(Total 4 marks)
6.A spinner has a red sector (R) and a yellow sector (Y).
The arrow is spun 1000 times,
(a)The results for the first 20 spins are shown below.
R R Y Y Y R Y Y R Y Y Y Y Y R Y R Y Y Y
Work out the relative frequency of a red after 20 spins.
Give your answer as a decimal.
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(2)
(b)The table shows the relative frequency of a red after different numbers of spins.
Number of spins / Relative frequencyof a red
50 / 0.42
100 / 0.36
200 / 0.34
500 / 0.3
1000 / 0.32
How many times was a red obtained after 200 spins?
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(2) (Total 4 marks)
Probability tree diagrams
1.Philip and Abdul run in different races.
The probability that Philip wins his race is 0.7 The probability that Abdul wins his race is 0.6
(a)Fill in the missing probabilities on the tree diagram.
(1)
(b)Calculate the probability that only one of the boys wins his race.
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(3)
(Total 4 marks)
2.Greg has four suits, one is striped and the other three are plain.
He also has ten shirts, four are white and the other six are coloured.
Greg chooses a suit at random and then chooses a shirt at random.
(a)Fill in the probabilities on the branches of the tree diagram.
(3)
(b)Calculate the probability that Greg chooses a plain suit and a coloured shirt.
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(2)
(Total 5 marks)
3.Danny has a biased coin.
The probability that the coin lands heads is .
Danny throws the coin twice.
(a)Fill in the probabilities on the tree diagram.
(2)
(b)Calculate the probability that Danny gets two heads.
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(2)
(Total 4 marks)
4.Jean enters an archery competition.
If it is raining the probability that she hits the target is 0.4.
If it is not raining the probability that she hits the target is 0.7
The probability that it rains on the day of the competition is 0.2
(a)Draw a fully labelled tree diagram showing all the probabilities.
(3)
(b)Calculate the probability that Jean hits the target with her first arrow in the competition.
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(3)
(Total 6 marks)
5.Shereen has two bags of marbles.
Bag A contains 3 red marbles and 4 green marbles.
Bag B contains 2 red marbles and 3 green marbles.
Shereen throws a fair six-sided dice.
If the dice lands on a six, she takes a marble at random from bag A.
If the dice lands on any other number, she takes a marble at random from bag B.
(a)Draw a fully labelled tree diagram showing the above information.
Mark the probabilities on the appropriate branches.
(3)
(b)Calculate the probability that a red marble is selected.
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(3)
(Total 6 marks)
Probability without replacements & listings
1.A bag contains 4 red, 3 yellow and 2 purple discs.
A disc is taken, at random, from the bag and is not replaced.
A second disc is then taken, at random, from the bag.
Calculate the probability that the two discs taken from the bag are
(a)both red,
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(2)
(b)different colours.
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(3)
(Total 5 marks)
2.A bucket contains tennis balls which are identical apart from their colour.
There are 5 yellow balls, 3 white balls and 2 green balls in the bucket.
Martina chooses two of the balls at random and without replacement.
What is the probability that the balls are the same colour?
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(Total 5 marks)
3.In a village of the pensioners have had a flu jab.
If a pensioner has had the flu jab the probability of catching flu is
If a pensioner has not had the flu jab the probability of catching flu is
(a)Calculate the probability that a pensioner, picked at random, from this village catches flu.
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(3)
(b)A statistician calculated that 120 pensioners from this village are expected to catch flu.
Calculate how many pensioners live in the village.
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(2)
(Total 5 marks)
4.Ingrid has 12 picture cards.
There are 2 apples, 3 pears and 7 bananas.
Ingrid chooses 2 cards at random.
Calculate the probability that both cards are the same.
You must show your working.
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(Total 5 marks)
5.A fair spinner has four equal sections.
The sections are coloured red (R), white (W), blue (B) and yellow (Y).
The arrow on the spinner is spun three times.
Calculate the probability that the arrow lands on the same colour at least twice.
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(Total 5 marks)
6.Some students decide to organise a day out.
They can only go on a Saturday or a Sunday.
of students choose a theme park.
The rest choose a water park.
of those choosing the theme park prefer Saturday.
of those choosing the water park prefer Sunday.
(a)One person is chosen at random.
Calculate the probability that this person prefers Saturday.
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(3)
(b)Of the students, 88 prefer Saturday.
How many students are there altogether?
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(2)
(Total 5 marks)
The RobertSmythSchool1