The Binomial Distribution - Worksheets, Questions and Revision

16 original exam-style questions - 11 pages of questions with a full mark scheme - free printable PDF.

Download PDFJump to mark scheme (page 12)
« Previous: The Normal DistributionNext: Quality Assurance and Control Charts »
Revision Library
revisionlibrary.co.uk
HIGHER

H07 The Binomial Distribution

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A random variable X can only be modelled by a binomial distribution if certain conditions apply to the trials being carried out. State the condition described in each part below.
(a)State a condition on the number of trials.(1)
(b)State a condition on the possible outcomes of each trial.(1)
(c)State a condition on the probability of success.(1)
(d)State a condition on how the trials relate to each other.(1)
(Total for Question 1 is 4 marks)
2
For each situation described below, state whether the random variable X can be modelled by a binomial distribution. If it cannot, give a brief reason why not.
(i)A fair coin is tossed 20 times. X is the number of heads obtained.(1)
(ii)Cards are drawn one at a time, without replacement, from a well-shuffled standard deck of 52 playing cards, and drawing continues until the first ace appears. X is the number of cards drawn.(1)
(iii)A biased six-sided dice, for which P(rolling a six) = 0.2 on each roll, is rolled 15 times. X is the number of sixes obtained.(1)
(iv)A bag contains 10 red counters and 10 blue counters. Six counters are drawn from the bag, one at a time, without replacement. X is the number of red counters drawn.(1)
(v)A multiple-choice test has 10 questions, each with exactly 4 options and only one correct answer. A student guesses the answer to every question at random, independently of their other guesses. X is the number of questions the student answers correctly.(1)
(Total for Question 2 is 5 marks)
3
A fair, ordinary six-sided dice is rolled 5 times. Let X be the number of times a 6 is obtained.
(a)Write down the value of n and the value of p for this binomial distribution.(2)
(b)Using the formula P(X = r) = nCr pr (1 - p)n-r, find P(X = 1), the probability that exactly one 6 is obtained.(3)
(Total for Question 3 is 5 marks)
4
A biased coin is designed so that P(Heads) = 0.3 on each throw. The coin is thrown 6 times. Let X be the number of heads obtained.
(a)Write down the value of n and the value of p.(2)
(b)Use the formula P(X = r) = nCr pr (1 - p)n-r to find P(X = 2).(3)
(c)Find P(X = 0).(2)
(Total for Question 4 is 7 marks)
5
A machine produces small bolts. Historical records show that 6% of the bolts produced are undersized. On a particular day, the machine produces 250 bolts. Let X be the number of undersized bolts among the 250, so that X ~ B(250, 0.06).
(a)State two assumptions, in the context of this machine, that must hold for X to be modelled in this way.(2)
(b)Calculate E(X), the mean number of undersized bolts expected among the 250 bolts produced that day.(2)
(c)After a repair to the machine, the mean number of undersized bolts per batch of 250 falls to 10. Find the new value of p, the probability that an individual bolt is undersized.(2)
(Total for Question 5 is 6 marks)
6
A fair coin is tossed 4 times. Let X be the number of heads obtained. The bar chart shows the probability distribution of X.
0 0.1 0.2 0.3 0.4 0.0625 0.25 0.375 0.25 0.0625 0 1 2 3 4 Number of heads, r Probability, P(X = r)
(a)Write down P(X = 3).(1)
(b)Find P(X ≤ 1), the probability of at most one head.(2)
(c)Show, using the formula P(X = r) = nCr pr (1 - p)n-r, that P(X = 2) = 0.375.(2)
(Total for Question 6 is 5 marks)
7
A factory finds that, on average, 5% of the light bulbs it produces are faulty. A random sample of 10 bulbs is selected for testing. Let X be the number of faulty bulbs in the sample.
(a)Write down the value of n and the value of p.(2)
(b)Find P(X = 2), the probability that exactly 2 bulbs in the sample are faulty.(3)
(c)Find the probability that at least one bulb in the sample is faulty.(3)
(Total for Question 7 is 8 marks)
8
A quality-control officer at a factory is investigating whether faulty coat hangers are still being produced at the same rate as before a machine setting was changed. Historically, 8% of coat hangers produced were faulty. She plans to take a random sample of 12 coat hangers from today's production and record X, the number of faulty hangers in the sample.
(a)State two conditions that must hold in this context for X to be modelled by the distribution B(12, 0.08).(2)
(b)Assuming X ~ B(12, 0.08), find P(X = 0), the probability that the sample contains no faulty hangers.(2)
(c)The officer collects her sample and finds that 3 of the 12 coat hangers are faulty. Using B(12, 0.08), find P(X = 3).(2)
(d)Using your answer to part (c), comment on whether the sample provides evidence that the proportion of faulty hangers has increased above 8%.(1)
(Total for Question 8 is 7 marks)
9
Historical data from a call centre shows that 25% of incoming calls are complaints. A random sample of 8 calls is selected. Let X be the number of complaint calls in the sample, so that X ~ B(8, 0.25).
(a)State what is meant by the values 8 and 0.25 in this context.(2)
(b)Find P(X ≤ 1), the probability that at most one of the 8 calls is a complaint.(4)
(c)Find P(X > 1), the probability that more than one of the 8 calls is a complaint.(2)
(Total for Question 9 is 8 marks)
10
A basketball player scores from any free-throw attempt independently, with probability 0.7 each time. In a training session she attempts 9 free throws. Let X be the number of free throws she scores, so that X ~ B(9, 0.7).
(a)Find P(X = 7), the probability that she scores exactly 7 of her 9 attempts.(3)
(b)Find P(X ≤ 7), the probability that she scores at most 7 of her 9 attempts, by first finding P(X = 8) and P(X = 9).(4)
(Total for Question 10 is 7 marks)
11
At a garden centre, 90% of tomato seeds germinate, independently of each other. A gardener plants 6 seeds in tray A and, separately, 6 seeds in tray B. Let X be the number of seeds that germinate in tray A, so that X ~ B(6, 0.9).
(a)Find P(X = 6), the probability that all 6 seeds in tray A germinate.(2)
(b)The number of seeds that germinate in tray B is independent of tray A. Find the probability that all 6 seeds germinate in both trays.(2)
(c)Find P(X = 5), the probability that exactly 5 of the 6 seeds in tray A germinate.(3)
(d)State, with a reason, whether it is more likely that exactly 5 seeds germinate in tray A or that exactly 6 seeds germinate in tray A.(1)
(Total for Question 11 is 8 marks)
12
A survey suggests that 40% of adults exercise regularly. A random sample of 10 adults is selected. Let X be the number who exercise regularly, so that X ~ B(10, 0.4).
(a)Find P(2 ≤ X ≤ 4), the probability that between 2 and 4 adults (inclusive) in the sample exercise regularly.(5)
(b)State two assumptions needed for this binomial model to be a reasonable one in this context.(2)
(Total for Question 12 is 7 marks)
13
A biased coin has P(Heads) = p on each throw. The coin is thrown 5 times. Let X be the number of heads obtained.
(a)Show that P(X = 2) = 10 p2 (1 - p)3, as required.(2)
(b)Given that p = 0.4, use your expression from part (a) to find the value of P(X = 2).(3)
(Total for Question 13 is 5 marks)
14
A fair coin is thrown n times. Let X be the number of heads obtained, so that X ~ B(n, 0.5). Given that P(X = 0) = 1/32.
(a)Find the value of n.(3)
(b)Using n = 5, find P(X ≥ 4), the probability of obtaining at least 4 heads, giving your answer as a fraction.(3)
(Total for Question 14 is 6 marks)
15
A university interview panel finds that, independently for each candidate, there is a probability of 0.65 that a candidate will be made an offer. In one session, 15 candidates are interviewed. Let X be the number of candidates made an offer, so that X ~ B(15, 0.65).
(a)Find P(X = 10), the probability that exactly 10 of the 15 candidates are made an offer.(3)
(b)State one assumption of this binomial model that may not be realistic in the context of interviewing candidates, and explain why.(2)
(Total for Question 15 is 5 marks)
16
A machine fills bottles of sauce. Let p be the probability that a randomly chosen bottle is underfilled. In a batch of 40 bottles, the mean number of underfilled bottles is 3.
(a)Show that p = 0.075.(2)
(b)Using p = 0.075, find the probability that a batch of 40 bottles contains exactly 2 underfilled bottles.(3)
(Total for Question 16 is 5 marks)
Mark scheme · H07 The Binomial Distribution

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16