Download 1. JLD Engineering is supplied a part from two different companies

1.
JLD Engineering is supplied a part from two different companies.
70% of these parts are supplied by Company A.
30% of these parts are supplied by Company C.
The quality of the parts supplied by Company A is 97% good and 3% bad.
The quality of the parts supplied by Company C is 95% good and 5% bad.
These data are represented on a probability tree diagram.
(a)
Complete the tree diagram.
(2)
A part is selected at random.
(b)
Work out the probability that the quality of the part is good.
............................................................
(2)
A part is selected at random.
(c)
Given that the part is good, work out the probability that it was supplied by Company A.
............................................................
(2)
(Total 6 marks)
Statistics Revision Pack 4 - Probabiilty
2.
Michael throws a stone at a target. He then throws a second stone at the target.
The probability he hits the target on his first throw is
1
5
If he hits the target on his first throw, the probability he will hit the target on his second throw is
1
3
If he misses the target on his first throw, the probability he will hit the target on his second throw is
(a)
1
4
Complete the probability tree diagram.
(3)
(b)
Work out the probability that Michael will hit the target on both throws.
.............................
(2)
(c)
Work out the probability that Michael will hit the target only once in the two throws.
.............................
(3)
Statistics Revision Pack 4 - Probabiilty
Gordon is going to throw five stones in turn at the target.
Gordon can hit the target with a probability of 0.8 with any one of these five stones.
(d)
Name the probability distribution that models the number of times he will hit the target in the five
throws.
.............................
(1)
(e)
Work out the probability that he will hit the target with only one of the five stones.
[You may use ( p  q) 5  p 5  5 p 4 q 10 p 3 q 2 10 p 2 q 3  5 pq 4  q 5 ]
.............................
(2)
(f)
Work out the most likely number of times he will hit the target.
[You may use ( p  q) 5  p 5  5 p 4 q 10 p 3 q 2 10 p 2 q 3  5 pq 4  q 5 ]
.............................
(2)
(Total 13 marks)
3.
The probability that Abena has a sickle cell gene is
The probability that Kofi has a sickle cell gene is
1
10
1
5
These probabilities are independent.
(Source: www.sicklecell.md)
Statistics Revision Pack 4 - Probabiilty
(a)
Complete the probability tree diagram.
Abena
Kofi
1
5
Has gene
Has gene
1
10
................
................
................
Does not
have gene
Has gene
Does not
have gene
................
Does not
have gene
(2)
(b)
Work out the probability that both Abena and Kofi have a sickle cell gene.
..............................
(2)
Abena and Kofi want to have 4 children together.
The probability that any child born to them will have sickle cell disease is
(c)
1
4
Work out the probability that at most one of their 4 children will have sickle cell disease.
[You may use (p + q)4 = p4 + 4p3q + 6p2q2 + 4pq3 + q4]
..............................
(4)
(Total 8 marks)
Statistics Revision Pack 4 - Probabiilty
4.
A travel agent organises several different tours to Germany, France and Switzerland.
Each tour goes to 1, 2 or 3 of these countries.
A total of 200 people went on these tours.
Of these
130 people went on a tour to Germany,
131 people went on a tour to France,
122 people went on a tour to Switzerland,
74 people went on a tour to Switzerland and France,
84 people went on a tour to France and Germany,
75 people went on a tour to Germany and Switzerland,
50 people went on a tour to all three countries.
(a)
Complete the Venn diagram for this information.
Germany
France
50
Switzerland
0
(3)
One of the 200 people is chosen at random.
(b)
(i)
Write down the probability that this person went on a tour to all three countries.
.....................................
(ii)
Write down the probability that this person did not go on a tour to France.
.....................................
(iii)
Work out the probability that this person went on a tour to Switzerland given that the person
also went to France.
.....................................
(4)
(Total 7 marks)
Statistics Revision Pack 4 - Probabiilty
5.
In a screening test a saliva sample is taken and tested for bacteria.
The result of a screening test can be either positive (T+) or negative (T–).
The probability of a person showing a positive (T+) result to the test is 0.1
(a)
Work out the probability that a person chosen at random will show a negative (T–) result to the
screening test.
............................
(1)
The probability that a person who gave a positive (T+) result to the screening test goes on to get tooth
decay (D+) within 18 months of the screening test was found to be 0.2
The probability that a person who gave a negative (T–) result to the screening test does not go on to get
tooth decay (D–) was found to be 0.9
(b)
Complete the tree diagram below.
Tooth decay
Outcome
Probability
of outcome
D+
T+D+
0.02
......
D–
T+D–
......
......
D+
T–D+
......
D–
T–D–
......
Screening test
result
0.2
T+
0.1
......
T–
0.9
(3)
(c)
Given that a person has tooth decay (D+), what is the probability that the person gave a positive
(T+) result in the screening test within the last 18 months?
............................
(2)
Statistics Revision Pack 4 - Probabiilty
(d)
What can you infer from this result about the usefulness of the screening test?
.....................................................................................................................................
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.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Data source: Dental Journal)
(Total 8 marks)
6.
A company runs holiday tours.
In 2005, 400 people made provisional bookings.
Of these 320 went on to confirm the booking.
In 2006 one person makes a provisional booking.
(a)
Write down an estimate of the probability that this person will go on to confirm the booking.
............................
(1)
(b)
Estimate the probability that the person will not go on to confirm the booking.
............................
(1)
(c)
The tour company has just had 4 people who made provisional bookings.
(i)
Calculate the probability that exactly 3 of these people will go on to confirm their booking.
You may use (p + q)4 = p4 + 4p3q + 6p2q2 + 4pq3 + q4.
Statistics Revision Pack 4 - Probabiilty
............................
(ii)
For these 4 provisional bookings, find which are the two most likely numbers of people who
go on to confirm their bookings. Show your working.
............................
(5)
(Total 7 marks)
7.
There are 30 people eating in a restaurant.
Each person chooses a first course of fish or meat or vegetarian, and a second course of cake or ice cream
or cheese.
The numbers of people eating each combination of first and second course are given in the table.
Choice of meals
Second course
First course
Cake
Ice cream
Cheese
Fish
4
4
3
Meat
5
3
4
Vegetarian
2
3
2
One of the 30 people is chosen at random.
(a)
Write down the probability that this person eats
(i)
fish for their first course and cake for their second course,
....................................
(ii)
ice cream for their second course, given their first course was vegetarian.
....................................
(2)
John says that people are equally likely to have fish or meat or vegetarian for their first course.
(b)
Does the data support this idea? Give a reason for your answer.
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(2)
(Total 4 marks)
Statistics Revision Pack 4 - Probabiilty
8.
Joan catches a bus, or cycles, or drives, to work.
The probability that Joan catches a bus to work is 0.4
The probability that she cycles to work is 0.5
The probability that she drives to work is 0.1
When Joan catches a bus, the probability of her being late is 0.2
When she cycles to work, the probability of her being late is 0.3
When she drives to work, the probability of her being late is 0.1
(a)
Complete the probability tree diagram below.
(2)
A day is chosen at random.
(b)
Find the probability that Joan is late for work.
...........................
(3)
Joan was late on Monday.
(c)
(i)
Work out the probability that she caught a bus.
...........................
Statistics Revision Pack 4 - Probabiilty
(ii)
Work out the probability that she did not catch a bus.
...........................
(3)
(Total 8 marks)
9.
Barry has two fair dice. Each dice has 6 faces. Barry rolls both dice.
He adds the numbers on the top of the two dice to get his total score.
(a)
Write down all the ways in which he can score 7.
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(2)
(b)
Write down the probability that he scores 7.
............................
(1)
Barry rolls both dice together 80 times.
(c)
Estimate how many times he scores 7.
............................
(1)
Megan throws darts at a target. She can hit the target with 7 out of 9 throws.
Megan throws three darts.
(d)
Assuming a binomial distribution, work out the probability that she hits the target exactly twice.
You may use (p + q)3 = p3 + 3p2q + 3pq2 + q3.
...................................
(2)
(e)
In order to assume a binomial distribution, what do you have to assume about the probability of
hitting the target with each of the three darts?
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(1)
Statistics Revision Pack 4 - Probabiilty
The assumption may not be realistic.
(f)
Write down one reason why.
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(1)
(Total 8 marks)
10. A health trust does a survey to see how many of its hospital beds have access to a radio and/or a
television. It finds that 21% have access to televisions only, 12% have access to radios only and 2% have
access to neither.
(a)
Complete the Venn diagram.
(2)
(b)
Find the probability that a bed chosen at random will have access to a radio or a television but not
both.
..........................................
(1)
(c)
Find the probability that a bed chosen at random will not have access to a television.
..........................................
(1)
(d)
Given that a patient is in a bed with access to a radio, what is the probability that the bed also has
access to a television?
..........................................
(2)
(Total 6 marks)
Statistics Revision Pack 4 - Probabiilty
11.
John is going on a 5-day holiday to Costa Packet. The travel brochure says that on average 5 out of every
7 days are sunny at Costa Packet. Each day’s weather is independent of the weather on all preceding
days.
(a)
(i)
Name the probability distribution that would model the number of sunny days at Costa
Packet.
..........................................
There are two values, n and p, that you need to use in this probability distribution.
(ii)
Write down the value of n and the value of p.
n = ...................................
p = ...................................
(2)
(b)
Calculate the probability that John has 2, or less, sunny days on his holiday?
You may use (p + q)5 = p5 + 5p4q + 10p3q2 + 10p2q3 + 5pq4 + q5.
..........................................
(3)
(c)
What is the most likely number of sunny days that John will get on his holiday?
Show your working.
..........................................
(2)
(Total 7 marks)
12.
Packets of nuts conform to the European minimum weight standard (e).
The packets in a supermarket are marked as 250 g e. This means that, on average, 1 out of every 40
packets (2.5%) weighs less than 241 g.
Assume that the weights of the packets of nuts are normally distributed, with mean 250 g.
(a)
Work out an estimate of the standard deviation of the weights of the packets.
................................ g
(2)
Statistics Revision Pack 4 - Probabiilty
The probabilities of buying 0, 1, 2, or 3 packets of nuts that do not conform to the European minimum
weight standard can be modelled by a binomial distribution.
Greta buys three packets of these nuts.
(b)
Work out the probability that she buys exactly two packets that do not conform to the European
minimum weight standard.
You may use (p + q)3 = p3 + 3p2q + 3pq2 + q3.
....................................
(3)
(c)
In order to use a binomial distribution as a model, an assumption is made about the probability that
the weight is less than 241g for each of the three packets.
Write down this assumption.
.....................................................................................................................................
.....................................................................................................................................
(1)
(Total 6 marks)
13. The probability that an electrical component is faulty is 0.05
(a)
Write down the probability that an electrical component is not faulty.
.....................................
(1)
(b)
Equal sized samples of the electrical components are selected at random. Give the name of a
distribution that can be used to model the number of faulty electrical components in the samples.
..............................................
(1)
Statistics Revision Pack 4 - Probabiilty
(c)
There are four electrical components in a sample.
(i)
Calculate the probability that exactly three of the electrical components are not faulty.
You may use (p + q)4 = p4 + 4p3q + 6p2q2 + 4pq3 + q4
.....................................
(ii)
Calculate the probability that more than one of the electrical components is faulty.
.....................................
(5)
(Total 7 marks)
Statistics Revision Pack 4 - Probabiilty