A LEVEL MATHEMATICS QUESTIONBANKS
BINOMIAL DISTRIBUTION
1. It is known that 80% of the seeds of particular flowers will germinate in the right conditions.
If a packet of 10 seeds is purchased, find the probability that:
a) at most two will fail to germinate.
[2]
b) exactly 8 will germinate.
[2]
c) Between 3 and 6 seeds inclusive will germinate
[3]
2.A student does no revision, and so always has to guess the answers in multiple-choice examinations.
a) If there are 50 questions, each with 4 possible answers, state the mean and variance of the
number of correct answers obtained by the student.
[3]
In another test, there are 10 questions, each with 5 possible answers. The student needs at least
7 correct answers to pass the test.
b) Find the probability that the student passes.
[2]
In a third multiple choice test, there are 20 questions, and the student has a probability p of selecting
the correct answer for each question.
The probability of the student getting more than half marks in this test is less than 0.01.
c) Use tables to explain why p must be less than 0.3, and hence deduce the minimum number of
possible answers for each question.
[4]
3.Whilst crossing a bridge the probability that a car will get a puncture is 0.00005. During a given day,
10,000 cars cross the bridge.
a) Find the probability that at least one car will get a puncture.
[3]
b) Find the probability that the fourth car to cross the bridge will be the first to get a puncture, giving
your answer to four significant figures
[3]
c) State one assumption you are making in your mathematical modelling.
[1]
4. On average, 5% of the items made on a production line are faulty.
a) Find the probability that a sample of 10 contains
i) No faulty items
[1]
ii) At least two faulty items
[2]
A sample of n items is taken from the production line.
b) If the probability of this sample containing at least one faulty item is greater than 99%, find the
smallest possible value of n. (You may use (0.95)88 =0.01096)
[5]
5. A newspaper article suggests that 15% of people are left-handed.
a) Use this information to suggest a suitable model for the number of left-handed people in a sample of n people.
[2]
As part of a statistics project, a student decides to test this model. He asks 10 students from his college
if they are left-handed.
b) Using the model you suggested in part a), find the probability that
i)None;
[1]
ii)Two or more
[2]
of the people from his sample are left-handed
The student finds that 8 of his sample are left-handed, and concludes that the 15% figure given must be wrong.
c) Using the tables provided or otherwise, explain why the student thought this
[3]
d) Give one reason why his conclusion is not necessarily correct.
[1]
6. Two alternative sampling processes are proposed for a quality control process:
Method 1:A sample of ten items is taken from a large batch and the batch rejected if 1 or more
defective item is found.
Method 2:A sample of twenty items is taken the whole batch is rejected if there are two or more
defective items in total.
The probability of a randomly selected item being defective is p.
a) Find, in terms of p, the probability of a batch being accepted if Method 1 is used
[2]
b) Show that the probability of a batch being accepted if Method 2 is used is given by:
(1 – p)19[ 1 + 19p]
[5]
It is decided to use both methods for quality control, with each method being used for half of the batches tested.
Given that p = 0.005,
c) find the probability that a batch that has been accepted was tested using method 1.
[8]
d) Comment on your answer
[1]
7.A bag contains 6 red and 3 black counters. I take a counter, note its colour and return it, and
repeat this until I have selected 5 counters.
a) Find the probability that I have selected
i) One red counter
[2]
ii) Two or more red counters
[3]
b) Given that I obtained at least one red counters, find the probability that I obtained two or more.
[4]
c) Without carrying out any further calculations, comment on how the probabilities in part a) would
change if I did not return the counter after each selection.
[2]
8.A cereal manufacturer claims that 80% of their boxes of cereal contain a free gift.
To test this a consumer group decides to take a random sample of 10 boxes and accept the claim
if 7 or more contain a gift.
Find the probability that:
a) the claim will be rejected even though it is true
[5]
b) the claim will be accepted if only 50% of the boxes contain a gift
[2]
c) Hence comment on this method for testing the claim, and suggest one way to improve the method.
[2]
9.A door-to-door salesman expects to make one sale for every 20 houses called on.
He earns £50 on every sale made.
a) If he calls on 20 houses, find the probability that he makes exactly one sale.
[3]
b) Find the number of houses he must call on to have an expected income of £500 per week.
[3]
c) If he calls on n houses, write down an expression, in terms of n, for the probability of him making
at least one sale.
[3]
d) Hence find the number of houses he must call on to be 90% certain of achieving at least one sale.
(You may use (0.95)46 = 0.099447)
[3]
10.In the manufacture of glass phials it is known that 15% will contain flaws. A random sample of ten
phials is chosen from a large batch:
a) Find the probability that a sample contains exactly one flawed phial
[2]
Fifty samples of size ten are taken
b) Find the probability that 20 of them contain exactly one flawed phial
[3]
c) Find the probability there are a total of 20 flawed phials among the 50 samples, giving your answer
in standard form to four significant figures
[3]
d) Explain why your answers to b) and c) are not identical.
[1]
11. Now that the main ITN weekday news bulletin has moved to 6:30pm a woman finds that she only gets home
from work in time to see the start of the programme on average twice in a working week of five days, each
day being equally likely.
a) Find the probability that in a given working week she will be home in time to see the start of the programme:
i)on 4 days
[2]
ii)on at least 4 days
[2]
iii)on 4 consecutive days but not the fifth.
[2]
b) Find the mean and standard deviation of the number of times she is home to see the start of the
programme during a period of six working weeks.
[4]
12. When a gymnast performs a routine, the probability she does so flawlessly is 0.65.
a) Find the probability that if she performs the routine 10 times, she will do so flawlessly on
i) 7 occasions
[2]
ii) At most 6 occasions
[4]
b) State one assumption that you have made in your mathematical modelling and comment on its validity.
[2]
13.a) State the conditions required for a binomial distribution to apply.
[2]
A student has collected data on the number of female children in the family from 100 households selected at random. He obtains the following results
Number of girls / 0 / 1 / 2 / 3 or moreNumber of households / 40 / 32 / 18 / 10
The student considers modelling this data using a binomial distribution
b) Explain why a binomial distribution is not suitable if the information in the table is all that is available.
[2]
A second student collects information on 80 families of 3 children
c) Suggest a suitable model for the number of girls per household, specifying the values of any parameters.
[2]
d) Using the model you have suggested, calculate the expected number of households in this sample having
0, 1, 2 and 3 girls.
[7]
The second student obtains the following data:
Number of girls / 0 / 1 / 23Number of households / 16 / 38 / 206
e) Without doing any further calculations, comment on the suitability of the model you suggested.
[2]
f) Suggest one way to refine your model
[1]
14.A student doing a statistics project decides to stand outside a shop at various times and note down the
sexes of the first ten people to enter it.
He decides to model the number of men per 10 people entering the shop using a binomial distribution, B(10, 0.5).
a) Comment on his choice of parameters for the binomial distribution
[2]
b) Comment on his use of the binomial distribution
[1]
The student finds that out of 200 people entering the shop, 60 were men. He decides to use this data to
refine his model.
c) Write down a suitable binomial model for the number of men per 10 people entering the shop.
[1]
Using your model,
d) find the probability that exactly 3 out of the next 10 people entering the shop are men.
[2]
e) find the probability that out of the next 20 samples of 10 people recorded by the student,
half contain exactly 3 men.
[4]
Page 1