Higher Revision – Probability
Grade: A*
1.
Hasna is using this fair six-sided spinner to play a game.
She spins it twice.
What is the probability she gets a 6 on exactly one of her spins?
3
4
.......................................
[4]
2.
Throw 3 sixes and win a DVD player!
DVD
(a)
Maria has one attempt to win a DVD player by throwing the three dice.
What is the probability that she wins a DVD player?
.....................................
[2]
(b)
Paul decides to have 5 attempts to win a DVD player.
What is the probability that Paul loses on his first four attempts
and then wins on his last attempt?
.................................
[3]
3.
At Hightown School there are 45 boys and 55 girls in Year 11
and 60 boys and 40 girls in Year 10.
A pupil is chosen at random from each year.
Work out the probability that one boy and one girl will be chosen.
......................................
[3]
South Wolds Comprehensive School
1
Grade A
4.
90% of students in a school are right-handed.
18% of students in the same school wear glasses.
These statements are independent.
(a)
Complete the tree diagram to show this information.
Wears glasses
Right-handed
0.9
Does not
wear glasses
Wears glasses
Left-handed
Does not
wear glasses
[2]
(b)
A student is chosen at random.
Calculate the probability that the student is left-handed
and wears glasses.
.....................................
[2]
5.
The Ace Driving School know from their records that:
•
The probability that a student passes the driving test at their first attempt is 0·7.
•
For a student failing at their first attempt, the probability of passing
at the second attempt is 0·8.
(a)
Complete the tree diagram.
First attempt
0·7
Second attempt
Pass
0·8
......
Pass
Fail
......
Fail
[1]
South Wolds Comprehensive School
2
(b)
Calculate the probability that a student
(i)
fails at their first attempt and passes at their second attempt,
..................................
[2]
(ii)
passes at either their first or second attempt.
..................................
[1]
6.
Anne drives to work.
She estimates that in November it rains on one morning out of three.
She also estimates that:
3
5
•
If it is raining the probability that she will be delayed is
•
If it is not raining the probability that she will be delayed is
1
5
Use the tree diagram to calculate the probability that on one November morning Anne will be
delayed.
Delayed
............
Raining
............
Not delayed
............
............
............
Delayed
............
Not delayed
Not
raining
....................................
[4]
South Wolds Comprehensive School
3
7.
The weather forecast for today says there is a 40% chance of rain.
The weather forecast for tomorrow says there is a 30% chance of rain.
Assume these events are independent.
(a)
Complete the tree diagram.
Tomorrow
Today
Rain
0·3
Rain
0·4
No rain
............
............
............
Rain
............
No rain
No rain
[2]
(b)
Calculate the probability of no rain on both days.
[2]
....................................
Grade B
8.
The probability of randomly choosing an ace from a pack of 52 playing cards is
1
13
Colin chooses a card at random from the pack, returns it to the pack, shuffles the pack and
chooses another card at random.
(a)
Complete the tree diagram to show the results of Colin’s choices.
First card
Second card
Ace
............
Ace
1
––
13
............
............
Not an
ace
............
Ace
............
Not an
ace
Not an
ace
[2]
South Wolds Comprehensive School
4
(b)
Calculate the probability that Colin chooses two aces.
....................................
[2]
9.
On her way to work, Yasmin goes through a set of traffic lights and over a level crossing.
The probability that she has to stop at the traffic lights is 0·4.
The probability that she has to stop at the level crossing is 0·3.
These probabilities are independent.
(a)
Complete the tree diagram to show this information.
Traffic lights
Level crossing
......
stop
......
not stop
......
stop
......
not stop
stop
......
......
not stop
[2]
(b)
Find the probability that, on her way to work, Yasmin stops at
both the traffic lights and the level crossing.
................................
[2]
10.
Neil travels to work in London either by bus or by tube.
The probability that he travels by tube is 0·7.
Whichever way he travels, the probability that he gets a seat is 0·8.
Sits
0.8
Tube
0.7
Stands
............
0.3
............
Sits
............
Stands
Bus
(a)
Complete the tree diagram.
[1]
South Wolds Comprehensive School
5
(b)
Calculate the probability that on a workday chosen at random,
Neil travels by tube and has to stand.
....................................
[2]
11.
Sports activities are held after school.
Zaneekia attends these classes on Monday and Wednesday.
She can choose one of badminton, dance or netball on each day.
The probability she chooses badminton is 0·5.
The probability she chooses dance is 0·4.
Assume Zaneekia’s choices are independent.
(a)
Complete the tree diagram below.
[1]
(b)
What is the probability Zaneekia will choose the same
sports activity on both days?
...................................
[4]
South Wolds Comprehensive School
6
12.
(a)
In November, a shop increased the price
of a television by 30%.
In the following January sale, the price of
the television was reduced by 30%.
Is the January sale price the same as the
price before the November increase?
Show calculations to explain your answer.
[3]
(b)
The shop offers a package where customers can choose
either a video player or a DVD player or a Hi-fi system
AND either a flat screen or a normal screen television.
Over a period of time, 35% of the customers choose a video player
and 60% choose a DVD player.
The rest choose a Hi-fi system.
Independently 75% of the customers choose flat screen televisions.
(i)
Complete this tree diagram.
0·75
Flat
Video
Normal
0·35
0·75
0·6
Flat
DVD
Normal
0·75
Flat
Hi-Fi
Normal
[2]
(ii)
All customers were entered into a prize draw.
What is the probability that the winner bought a Hi-Fi
system with a normal screen television?.
[2]
(c)
The size of a flat screen television is given as
86 cm correct to the nearest centimetre.
What is the maximum and the minimum possible
screen size?
maximum size = ................................... cm
minimum size = ................................... cm
[2]
South Wolds Comprehensive School
7
Grade C
13.
A bag contains red, blue and green counters.
A counter is drawn at random from the bag.
The probability that it is red is 0·4.
The probability that it is blue is 0·25.
(a)
What is the probability that it is green?
...................................
[2]
(b)
There are 80 counters in the bag.
How many of them are blue?
..................................
[2]
14.
In a children’s ball-pool there are green, yellow, orange and blue balls.
Ramy picks a ball up at random.
This table shows the probabilities of obtaining each colour.
Colour
Probability
Green
0 ⋅2
Yellow
0 ⋅3
Orange
0 ⋅4
Blue
(a)
Complete the table.
[2]
(b)
There are 1000 balls in the ball-pool.
How many of them are yellow?
....................................
[2]
South Wolds Comprehensive School
8
15.
Maggie has a box of chocolates.
It contains milk, plain and white chocolates.
Maggie chooses a chocolate at random.
The probability of choosing a milk chocolate is
(a)
3
.
8
There are 40 chocolates in the box.
How many are milk chocolate?
.....................................
[2]
(b)
The probability of choosing a plain chocolate is
1
.
2
What is the probability of choosing a white chocolate?
....................................
[1]
16.
Geoff picks 40 tomatoes and weighs them.
The results are summarised in the table below.
(a)
Mass (m grams)
Frequency
Mid-interval value
0 ≤ m < 25
6
12·5
25 ≤ m < 50
10
37·5
50 ≤ m < 75
16
62·5
75 ≤ m < 100
8
87·5
Calculate an estimate of the mean mass of the tomatoes.
................................. g
[3]
(b)
Geoff takes one of these tomatoes at random.
What is the probability that it weighs at least 50 grams?
....................................
[1]
South Wolds Comprehensive School
9
17.
Catherine recorded the number of minutes her train was late.
Her results for 80 journeys are summarised in the table below.
(a)
Minutes late (t)
Frequency
0 ≤ t < 10
34
10 ≤ t < 20
25
20 ≤ t < 30
12
30 ≤ t < 40
7
40 ≤ t < 50
2
Calculate an estimate of the mean number of minutes that her train was late for these 80
journeys.
....................... minutes
[4]
(b)
One of these journeys is picked at random.
What is the probability that the train was less than 20 minutes late?
.....................................
[2]
Jana often has to cross a railway at a level crossing.
O
CR
CR
Y
WA
Y
WA
O
SS
IN
G
IL
IL
(a)
G
RA
RA
SS
IN
18.
Over a period of time Jana kept a record of whether or not she had to stop at the crossing.
Frequency
Stop
13
Not stop
27
Jana uses the crossing 500 times in a year.
About how many times will she expect to stop?
.......................................
[2]
South Wolds Comprehensive School
10