Department of Electrical and Electronic Engineering Mr R A JUGURNAUTH

PROBABILITY PROBLEM SHEET ONE: Basic Statistics, Sets and Combinations

Question 1

Find the mean ( xμ = ), mean square value ( 2x ), the variance ( 2σ ), and the standard deviation (σ ) for the

following data: { } . Confirm that 10 {1,6,9, 4, 2,6, 5,8}N

i ix −= = − − − 2 2x 2σ μ= + for this particular case. Show that

2 2x 2σ μ= + holds for any set of data. Finally, a standard deviation of 2 about a mean of 4 represents a different effect than a standard deviation of 2 about a mean of 20. So we define the coefficient of variation as

100%V σμ

= × . What is V in this case? Does all the data lie less that two standard deviations from the mean?

Question 2 Standard units tell the number of standard deviations a given value lies above or below the mean of a

population/data set. Let the value of x in standard units, denoted by z, be defined as: xz μσ−

= . It can be used to

compare values from different samples or populations. Ten students take two tests - Maths and Physics. In the Maths test Mary gets 67% and 60% in the Physics test. Given that one test may be considered ‘harder’ than the other, use standard units to determine in which test did Mary get a ‘higher’ score. The % scores for the ten students for the two tests is given below. Maths Marks: {67 74 44 93 46 41 84 52 20 1} Physics Marks: {60 35 81 0 13 20 19 5 27 19} Question 3 Which set of data goes with which histogram? The means are 0.536 and 4.096; the standard deviations are 0.2630 and 1.728. Which goes with which?

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12HISTOGRAM OF X

0 10 20 30 40 50 60 70 80 90 100

-1

0

1

2

3

4

5

6

7

8

9

D1

-1 0 1 2 3 4 5 6 7 8 90

5

10

15

20

25HISTOGRAM OF X

H2

H1

D2

Question 4 Let { }1,2,3,4,U be the universal set and let = K { } { } { } {1,2,3,4 , 3,4,5,6,7 , 6,7,8,9 and 2,4,6,A B C E= = = = K} . Find the following sets: (i) , , and A B E ; (ii) \ , \ , \ , \ and \A B B C B A C B C E ; (iii) ; and A B B C⊕ ⊕(iv) From the above results confirm that ( \ ) ( \ ) and ( \ ) ( \ )A B A B B A B C B C C B⊕ = ∪ ⊕ = ∪ .

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Question 5 Let be the universal set and let { , , , ,U a b c d e= } { } { }, , , , ,A a b d B b d e= = . Find the following sets: (i) A B∪ ; (ii) ; B A∩(iii) B ; (iv) ; \B A(v) A B∩ ; (vi) A B∪ ; (vii) A B∩ ; (viii) \B A ; (ix) ( )A B∩ ;

(x) ( )A B∪ . Question 6 (i) Using a Venn diagram, prove the OR rule for probabilities for two events – i.e. prove

, which is the same as ( ) ( ) ( ) (P A B P A P B P A B∪ = + − ∩ ) ( or ) ( ) ( ) ( and )P A B P A P B P A B= + − ; (ii) Using a Venn diagram prove the OR rule for probabilities for three events – i.e. prove

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )P A B C P A P B P C P A B P A C P B C P A B C∪ ∪ = + + − ∩ + ∩ + ∩ + ∩ ∩ ; (iii) What is the general rule for n events? (iv) Using the above results, what is the probability that a bridge hand (13 cards) contains 4 aces or 4 kings? (v) Using the above results, what is the probability that a poker hand (5 cards) contains all spades or all hearts? Question 7 I deal a hand of 5 cards from 52 playing cards. (i) What is the probability that all cards are clubs? (ii) What is the probability that at least one is a King? (iii) What is the probability that we have two pairs? Question 8 Consider the following experiments, and answer the questions. Experiment One: Select a ball from an urn containing balls numbered 1 to 50. Experiment Two: Pick a number at random between zero and one Experiment Three: Pick two numbers X and Y at random between zero and one. Experiment Four: Pick a number X at random between zero and one, then pick a number Y at random between zero and X. Experiment Five: Select a ball from an urn containing balls numbered 1 to 4. Suppose that balls 1 and 2 are black and that balls 3 and 4 are white. Note the number and the colour of the ball that you select.

(i) Give expressions for the sample spaces for the first five experiments. (ii) Draw the sample spaces for the first four experiments. (iii) Give expressions for the following events in experiment one: (a) E1 = Even ball selected; b) E2 = Odd

ball selected; c) E3 = Prime number ball selected; d) Give some relationships between the sets E1 , E2 and E3.

(iv) Consider experiment four. Sketch the event E4 where the sum of X plus Y is greater than 0.5. Question 9

(i) A bag contains one red and three white balls. If two are picked at random, what is the probability that both are white? Answer this question in four different ways – using either an exhaustive proof; combinations; permutations; or individual probabilities.

(ii) Repeat four proofs for one red and three whites. Choose three. What is probability that two are white? Question 10 A bag contains 3 white balls, 4 black balls and 5 red balls. If I choose four balls at random find the probability that (i) I have zero reds; (ii) I have one red; (iii) I have two reds; (iv) I have at least three reds; (v) Repeat (iv) a different way. Question 11 Seven husbands (H1 to H7) are separated from their seven wives (W1to W7). They are then paired at random.

(i) What is the probability that H1 is matched with his wife?; (ii) What is the probability that H2, H5 and H7 are all correctly matched?; (iii) What is the probability that at least one husband is matched to his wife? (iv) What is the probability that no husband is matched to his wife?

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Department of Electrical and Electronic Engineering Mr R A JUGURNAUTH

PROBABILITY PROBLEM SHEET TWO: Conditional and Bayesian

Probability

Question 1 Consider three mutually exclusive and exhaustive events: 1 2 3, and A A A . Let there be another event, . With the aid of Venn diagrams, give a simple an intuitive derivation of Bayes’ Law to expand . Also explain the purpose of Bayes’ Law.

B

1( | )P A B

Question 2 If two events are such that the occurrence (or non-occurrence) of one, does not affect the likelihood of the other event, the events are called independent. The mathematical definition is:

and are independent if ( ) ( ) ( )A B P A B P∩ = A P B What does this say about the conditional probabilities? Question 3

(i) Prove that ( | ) 1 ( | ) P A B P A B= − (ii) Prove that ( ) ( ) ( | ) ( | ( )P A B C P A P B A P C A B∩ ∩ = ∩(iii) Generalise (ii) for , and others. ( )P A B C D∩ ∩ ∩

Question 4 Find the probability that a poker hand (of five cards) contains two Kings if you know that it already contains exactly one Queen. Question 5 Suppose a box contains five white and seven red balls. Two balls are drawn without replacement.

(i) What is the probability that both balls are white? (ii) What is the probability that the second ball is red? (iii) What is the probability that the second ball is white? (iv) What is the probability that given the second ball is red, the first ball was also red? (v) What is the probability that given the second ball is red, the first ball was white?

Question 6 The problem of colour blindness is 0.02 for a man and 0.001 for a woman. Find the probability that a person picked at random is colour blind if the population is 53% men. Question 7 An insurance company believes that 30% of drivers are careless, and that the probability of a driver having an accident in any one year is 0.4 for a careless driver and 0.2 for a careful driver. Find the probability that a driver will have an accident next year given that she has had an accident this year.

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R JUGURNAUTH UOM

ELEC 2001Y – Analytical Techniques – Probability Problem Sheet 3 – Discrete Random Variables (1) A and B play a series of tennis matches. The probability that A wins any single match in the series is 0.6. The winner of the series is

the first player to win either two matches in succession or a total of three matches. Show that the probability (a) that the series lasts exactly two matches is 0.52 (b) that the series lasts exactly three matches is 0.24 Calculate the probability that the series lasts exactly four matches. Hence, or otherwise, show that the probability the series lasts five matches is 0.1152. Calculate the expectation of n, the number of matches in the series. The prize-money involved depends on n and is shown in the table below.

n 2 3 4 or 5 Prize-money Rs 1000 Rs 1240 Rs 1510

Tickets are sold, each of which entitles the purchaser to see the whole series of matches. Given that each ticket costs Rs 5, calculate the number of tickets which must be sold to cover the expected value of the prize-money.

(2) A box contains five discs, labelled 1, 2, 4, 5, and 6. In a game, a player draws a disc at random, replaces it and then draws again. The player’s score is the sum of the numbers on the two discs drawn.

Construct a table showing the 11 possible scores and their probabilities. Find the expected score. In a social club, this game is played and the prize is Rs 10 for each point scored. The players pay Rs 75 each time they play. Find the expected profit to the club after 250 games have been played.

(3) Ten playing cards, two of which are Aces, are lying face down on a table. A player turns the cards over, one by one, in a random

order, and the Nth card turned over is the first Ace found ( 91 ≤≤ N ). Calculate the probabilities P (N = 2) and P (N = 9). A player pays a stake of Rs 10 at finding an Ace. If the first card he turns is an Ace, he gets his stake of Rs 10 back, and in addition he gets a prize of Rs 30. If he fails to find an Ace with his first card, he turns a second card, and if this is an Ace, his Rs 10 stake is returned, but otherwise he gets nothing at all and his stake is lost. Calculate a player’s expectation, stating whether it is a gain or a loss.

(4) A test consists of five multiple choice questions to each of which three answers are given, only one of which is correct. For each

correct answer, a candidate gets 2 marks, but loses 1 mark for each incorrect answer. A particular candidate answers each question purely by guesswork. Draw up a table to show all the possible total marks from -5 to 10 and the probability associated with each mark for the candidate. Use the table to show that the expected mark is zero, and find the variance of this distribution.

(5) Ten ordinary dice are thrown and the number of sixes showing as a result is counted. Find the probability that, in one throw of the ten

dice: (a) exactly 3 sixes will show (b) more than 3 sixes will show In a game, a player pays a stake of $1 to throw the ten dice. The player receives nothing if fewer than 2 sixes show as a result of the throw, receives $1 if 2 sixes show, receives $2 if 3 sixes show, and receives $k if more than 3 sixes show. Find the value of k for which the expectation of the amount the player receives is equal to the player’s stake. In a second game, a player pays a stake to throw n ordinary dice. The player receives $ ⎟

⎠

⎞⎜⎝

⎛ x21 if x sixes show, where x = 0, 1, 2, …, n. Show that the expectation of the amount the

player receives is $ ⎟⎠

⎞⎜⎝

⎛ n121 .

(6) An electrical circuit contains 5 components, one of which is faulty. To isolate the fault, the components are tested one by one until the

faulty one is found. The random variable X denotes the number of tests requires to locate the fault. The test of the faulty component itself is always included, so that X takes values from 1 to 5 inclusive. Find the expectation and variance of X. The cost C (in suitable units) of locating a fault depends in part on the number of tests required and is given by C = 5 + 2X. Find the expectation and variance of C. One hundred circuits of the above types are to be tested. Find the probability that the average cost will exceed 11.5 units.

(7) A coin and a six-faced die are thrown simultaneously. The random variable X is defined as follows: If the coin shows a head, then X

is the score on the die, and if the coin shows a tail, then X is twice the score on the die. Find the expected value, µ, of X and show that P(X < µ) = 7/12. Show that Var(X) = 497/48. The experiment is repeated and the sum of the two values obtained for X is denoted by Y. Find P(Y = 4) and E(Y).

R JUGURNAUTH UOM

ELEC 2001Y – Analytical Techniques – Probability Problem Sheet 4 – Discrete Random Variables (1) An instructor who taught two sections of statistics last term, the first with 20 students and the second with 30, decided to

assign a term project. After all projects had been turned in, the instructor randomly ordered them for grading. Consider the first 15 graded projects.

(a) What is the probability that at least 10 of these are from the second section? (b) What is the probability that at least 10 of these are from the same section? (c) What are the mean value and standard deviation of the number among these 15 that are from the second section?

(d) What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section?

(2) A manufacturer sells a certain article in batches of 5000. By agreement with a customer, the following method of inspection is

adopted: a sample of 100 items is drawn at random from each batch and inspected. If the sample contains 4 or fewer defective items, then the batch is accepted by the customer. If more than 4 defective items are found, every item in the batch is inspected. If inspection costs are 75 cents per hundred articles, and the manufacturer normally produces 2% of defective articles, find the average inspection cost per batch.

(3) In one part of a country, one person in 80 has blood of Type P. A random sample of 150 blood donors is chosen from that part

of the country. Let X represent the number of donors in the sample having blood of Type P. (a) State the distribution of X. Find the parameter of the Poisson distribution which can be used as an approximation. Give a

reason why a Poisson distribution is appropriate. (b) Using the Poisson distribution, calculate the probability that in the sample of 150 donors, at least two have blood of Type

P. (c) A hospital urgently requires blood of Type P. How large a random sample of donors must be taken in order that the

probability of finding at least one donor of Type P must be 0.99 or more. (4) A randomly chosen doctor in general practice sees, on average, one case of a broken nose per year and each case is

independent of other similar cases. (a) Regarding a month as a twelfth part of a year, (i) show that the probability that, between them, three such doctors see no cases of a broken nose in a period of one

month is 0.779, correct to three significant figures, (ii) find the variance of the number of cases seen by three such doctors in a period of six months (b) Find the probability that, between them, three such doctors see at least three cases in one year (c) Find the probability that, of three such doctors, one sees three cases and the other two see no cases in one year.

(5) Before starting to play the game ‘Snake and Ladders’ each player throws an ordinary unbiased die until a six is obtained. The

number of throws before a player starts is the random variable Y, where Y takes the values 1, 2, 3, … (a) Name the probability distribution of Y. (b) Find Var(Y). (c) Two people play Snakes and Ladders. Calculate the probability that they will each need at least five throws before

starting. (6) During a weekday, heavy lorries pass a point P on a village high street independently and at random times. The mean rate for

westward traveling lorries is two in any 30-minute period, and for eastward traveling lorries is three in any 30-minute period. Find the probability

(a) that there will be no lorries passing P in a given 10-minute period (b) that at least one lorry from each direction will pass P in a given 10-minute period (c) that there will be exactly four lorries passing P in a given 20-minute period.

ELEC 2001Y – Analytical Techniques – Probability Problem Sheet 5 – Continuous Random Variables (1) Machine A, used for filling bags with ground coffee, can be set to dispense any required mean weight of coffee per bag. At

any setting, the weight of coffee in a bag can be modeled by a normal distribution with a standard deviation of 1.95 g. (a) If the machine is set to dispense a mean weight of 128 g of coffee per bag, calculate the percentage of bags that contain

less than 125 g. (b) To meet an official regulation, the setting on a machine must be adjusted so that no more than 1 % of bags contain less

than 125 g. (i) Calculate the smallest mean weight to which machine A should be set to meet he regulation. (ii) Machine B will only just meet the regulation when it is set to dispense a mean weight of 128.5 g. Assuming that the

weight of coffee in a bag filled by machine B can be modeled by a normal distribution, calculate the standard deviation of this distribution.

(2) The weights of grade A oranges are normally distributed with mean 200 g and standard deviation 12 g. Determine, correct to 2

significant figures, the probability that, (a) a grade A orange weighs more than 190 g but less than 210 g (b) a sample of 4 grade A oranges weighs more than 820 g The weights of grade B oranges are normally distributed with mean 175 g and standard deviation 9 g. Determine, correct to 2

significant figures, the probability that, (c) a grade B orange weighs less than a grade A orange (d) a sample of 8 grade B oranges weighs more than a sample of 7 grade A oranges.

(3) It is estimated that, on average, one match in five in the Football League is a draw, and that one match in two is a home win. (a) Twelve matches are selected at random. Calculate the probabilities that the number of drawn matches is (i) exactly three (ii) at least four (b) Ninety matches are selected at random. Use a suitable approximation to calculate the probability that between 13 and 20

(inclusive) of the matches are drawn. (c) Twenty matches are selected at random. The random variables D and H are the numbers of drawn matches and home

wins, respectively, in these matches. State, with a reason, which of D and H can be better approximated by a normal variable.

(4) Out of a random sample of 1000 French people interviewed in 1996, 410 supported a single European currency. (a) Calculate an approximate 99 % confidence interval for the population proportion, p, of French people who supported a

single European currency. (b) Estimate the size of a sample that would have provided a 99 % confidence interval of width 0.04 for p. (c) Justify why your answer to (b) is only an estimate. (5) Each of a random sample of 50 one-rupee coins was weighted and their masses, x grams, are summarized by , 51.474=∑ x 8276.45032 =∑ x

(a) Use an unbiased estimate of variance to calculate an approximate 90 % confidence interval for the mean mass (in grams) of all one-rupee coins, giving the end-values of the interval to two decimal places.

(b) Estimate the size of a random sample of one-rupee coins that would be required to give a 95 % confidence interval whose width is half that of the interval calculated in (a).

R JUGURNAUTH UOM

Question 4 (a) Let S be the set of all possible outcomes of an experiment. Assume A1, A2

and A3 are mutually exclusive and exhaustive events within the sample space S. Let D be any other event. Illustrate this scenario with the aid of a Venn diagram, and thus explain in set notation (using and ∩ ∪ symbols) the meanings of the words (i) mutually exclusive and (ii) exhaustive in the context of the experiment. [3 marks]

(b) With the aid of a Venn diagram explain why for any two events A and B,

P(A|B) = P(A∩B)/P(B) and P(B|A) = P(B∩A)/P(A). [3 marks] (c) (i) If A and B are two events such that 0)( ≠AP and , prove

that if 0)( ≠BP

)()( APBAP > then )()( BPABP > . [3 marks] (ii) If A , B and are three events such that and

, show that C 0)( ≠CP

0)( ≠∩B CP )()()()( CPCBPCBAPCBAP ∩=∩∩ . [2 marks]

(d) For A1, A2, A3 and D in part (a), since

, then starting with the result in part (b), prove Bayes’ Rule:

1 2( ) ( ) ( ) ( )P D P D A P D A P D A= ∩ + ∩ + ∩ 3

1 1

11 1 2 2 3 3

( ) ( | )( | )( | ) ( ) ( | ) ( ) ( | ) ( )

P A P D AP A DP D A P A P D A P A P D A P A

=+ +

.

[6 marks]

(e) Let 100 bits/sec be received from transmitter A1 with error rate =1/5; let 200 bits/sec be received from transmitter A2 with error rate =1/10; let 400 bits/sec be received from transmitter A3 with error rate =1/40. Let all three bit streams be randomly multiplexed together, and let one bit be selected at random from the composite signal. Let D represent the event that this bit is in error. By using the results from part (d):

(i) What is the probability (P(D)) that the selected bit is in error?

[4 marks] (ii) If the selected bit is in error, what is the probability that it is from

transmitter A1? [4 marks]

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Question 5 (a) Let the voltage V(t) represent some parameter in a communications

system. Let it also be a continuous random variable with pdf:

, for 3( )

0, for 3Va

pλ λ

λλ

⎧ ≤⎪= ⎨>⎪⎩

.

(i) Sketch ( )Vp λ , and from this explain why the value of the constant

‘a’ must equal 1/9; [2 marks] (ii) Determine the mean, Vμ , the variance, 2

Vσ , and the power ( 2V ) of V(t); [4 marks]

(iii) Calculate the probability that the voltage has a value 1 ( )V t≤ ≤ 2 .

[2 marks]

(b) A shop sells rose plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95. (i) Calculate the number of plants per box. [2 marks] (ii) Calculate the probability that a box contains exactly 12 plants which

produce yellow flowers. [2 marks] (iii) Another shop sells rose plants in boxes of 100. The shop’s

advertisement states that the probability of any rose plant producing a pink flower is 0.3. Use a suitable approximation to calculate the probability that a box contains fewer than 35 plants which produce pink flowers. [3 marks]

(c) The probability distribution function for a Poisson random variable with parameter λ is given by

!)(

kekXP

kλλ−== ,.........1,0=k

(i) Under what conditions does this probability distribution function

arise. [2 marks] (ii) For a Poisson random variable X with parameter λ show

that λμ =X and . [8 marks] λσ =2X

Page 5 of 7

Question 6 (a) A packet contains six biscuits, each of which is individually wrapped.

The mass of a biscuit can be taken to be normally distributed with mean 70 g and standard deviation 4 g. The mass of the individual wrapping of a biscuit is normally distributed with mean 10 g and standard deviation 1 g. The mass of the outer packaging is normally distributed with mean 30 g and standard deviation 3 g. Assuming that the masses of the biscuits, wrappings and packaging are independent, calculate the probability that the total mass of a randomly chosen packet and its contents lies between 500 g and 520 g. [6 marks]

(b) An experiment carried out to compare the bond strength of modified

mortar to that of unmodified mortar resulted in x = 18.12 kgf/cm2 for the modified mortar (sample size = 40) and 1n y = 16.87 kgf/cm2 for the unmodified mortar (sample size = 32). Assume that the bond strength distributions are both normal.

2n

(i) Assuming that standard deviations 1σ = 1.6 and 2σ = 1.4 for the

modified mortar and the unmodified mortar respectively, test at level 0.01 if the bond strength for modified mortar is greater than that for the unmodified mortar. [4 marks]

(ii) Calculate the probability of a type II error for the test of part (i)

when mean difference between the bond strengths of modified mortar and unmodified mortar is actually 21 μμ − = 1. [4 marks]

(iii) How would the analysis and conclusions of part (i) change if the

population standard deviations 1σ and 2σ were unknown, but instead the sample standard deviations = 1.6 and = 1.4 were known? [4 marks]

1s 2s

(c) Each helmet in a random sample of 42 helmets was subjected to a certain

impact test and 27 showed damage.

(i) Calculate a 99 % Confidence Interval of the proportion of all helmets that will show damage when tested in this manner.

[4 marks] (ii) What sample size would be required for the width of a 99 %

Confidence Interval to be at most 0.10? [3 marks]

END OF QUESTION PAPER

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