5.01a Permutations and combinations: evaluate probabilities

4 In a population colonizing an island 40\% of the first generation (parents) have brown eyes, \(40 \%\) have blue eyes and \(20 \%\) have green eyes. Offspring eye colour is determined according to the following rules. \section*{Eye colours of parents Eye colour of offspring} (1) both brown
(2) one brown and one blue \(50 \%\) brown and \(50 \%\) blue
(3) one brown and one green blue
(4) both blue \(25 \%\) brown, \(50 \%\) blue and \(25 \%\) green
(5) one blue and one green 50\% blue and \(50 \%\) green
(6) both green green

1. Give an efficient rule for using 1-digit random numbers to simulate the eye colour of a parent randomly selected from the colonizing population.
2. Give an efficient rule for using 1-digit random numbers to simulate the eye colour of offspring born of parents both of whom have blue eyes. The table in your answer book shows an incomplete simulation in which parent eye colours have been randomly selected, but in which offspring eye colours remain to be determined or simulated.
3. Complete the table using the given random numbers where needed. (You will need your own rules for cases \(( 2 )\) and 5 .)
   Each time you use a random number, explain how you decide which eye colour for the offspring. \(\square\)

4 A ski-lift gondola can carry 4 people. The weight restriction sign in the gondola says "4 people - 325 kg ". The table models the distribution of weights of people using the gondola.

1. Give an efficient rule for using 2-digit random numbers to simulate the weight of a person entering the gondola.
2. Give a reason for using 2-digit rather than 1-digit random numbers in these circumstances.
3. Using the random numbers given in your answer book, simulate the weights of four people entering the gondola, and hence give its simulated load.
4. Using the random numbers given in your answer book, repeat your simulation 9 further times. Hence estimate the probability of the load of a fully-laden gondola exceeding 325 kg .
5. What in reality might affect the pattern of loading of a gondola which is not modelled by your simulation?

5 Five gifts are to be distributed among five people, A, B, C, D and E. The gifts are labelled from 1 to 5. Each gift is allocated randomly to one of the five people. A person can receive more than one gift.

1. Use one-digit random numbers to simulate this process. One-digit random numbers are provided in your answer book. Explain how your simulation works. Produce a table, showing how many gifts each person receives.
2. Carry out four more simulations showing, in each case, how many gifts each person receives.
3. Use your simulation to estimate the probabilities of a person receiving \(0,1,2,3,4\) and 5 gifts.
4. Describe what you would have to do differently if there were six people and six gifts.

5 A chairlift for a ski slope has 160 4-person chairs. At any one time half of the chairs are going up and half are coming down empty. An observer watches the loading of the chairs during a moderately busy period, and concludes that the number of occupants per 'up' chair has the following probability distribution.

1. Give a rule for using 1-digit random numbers to simulate the number of occupants of an up chair in a moderately busy period.
2. Use the 10 random digits provided to simulate the number of occupants in 10 up chairs. The observer estimates that, at all times, on average \(20 \%\) of chairlift users are children.
3. Give an efficient rule for using 1-digit random numbers to simulate whether an occupant of an up chair is a child or an adult.
4. Use the random digits provided to simulate how many of the occupants of the 10 up chairs are children, and how many are adults. There are more random digits than you will need.
5. Use your results from part (iv) to estimate how many children and how many adults are on the chairlift (ie on the 80 up chairs) at any instant during a moderately busy period. In a very busy period the number of occupants of an up chair has the following probability distribution.

   number of occupants	0	1	2	3	4
   probability	\(\frac { 1 } { 13 }\)	\(\frac { 1 } { 13 }\)	\(\frac { 3 } { 13 }\)	\(\frac { 3 } { 13 }\)	\(\frac { 5 } { 13 }\)

6. Give an efficient rule for using 2-digit random numbers to simulate the number of occupants of an up chair in a very busy period.
7. Use the 2-digit random numbers provided to simulate the number of occupants in 5 up chairs. There are more random numbers provided than you will need.
8. Simulate how many of the occupants of the 5 up chairs are children and how many are adults, and thus estimate how many children and how many adults are on the chairlift at any instant during a very busy period.
9. Discuss the relative merits of simulating using a sample of 10 chairs as against simulating using a sample of 5 chairs.

5 A computer store has a stock of 10 laptops to lend to customers while their machines are being repaired. On any particular day the number of laptop loans requested follows the distribution given in Table 5.1. \begin{table}[h] \end{table}

1. Give an efficient rule for using two-digit random numbers to simulate the daily number of requests for laptop loans.
2. Use two-digit random numbers from the list below to simulate the number of loans requested on each of ten successive days. Random numbers: \(23,02,57,80,31,72,92,78,04,07\) The number of laptops returned from loan each day is modelled by the distribution given in Table 5.2, independently of the number on loan (which is always at least 5 ). \begin{table}[h]

   Number returned	0	1	2	3
   Probability	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 3 }\)

   \captionsetup{labelformat=empty} \caption{Table 5.2}
   \end{table}
3. Give an efficient rule for using two-digit random numbers to simulate the daily number of laptop returns.
4. Use two-digit random numbers from the list below to simulate the number of returns on each of ten successive days. Random numbers: \(32,98,01,32,14,21,32,71,82,54,47\) At the end of day 0 there are 7 laptops out on loan and 3 in stock. Each day returns are made in the morning and loans go out in the afternoon. If there is no laptop available the customer is disappointed and never gets a loaned laptop.
5. Use your simulated numbers of requests and returns to simulate what happens over the next 10 days. For each day record the day number, the number of laptops in stock at the end of the day, and the number of customers that have to be disappointed.
   [0pt] [3] To try to avoid disappointing customers, if the number of laptops in stock at the end of a day is 2 or fewer, the store sends out e-mails to customers with loaned laptops asking for early return if possible. This changes the return distribution for the next day to that given in Table 5.3. \begin{table}[h]

   Number returned	0	1	2	3	4
   Probability	0.1	0.1	0.4	0.2	0.2

   \captionsetup{labelformat=empty} \caption{Table 5.3}
   \end{table}
6. Simulate the 10 days again, but using this new policy. Use the requests you produced in part (ii). Use the random numbers given in part (iv) to simulate returns, but use either the distribution given in Table 5.2 or that given in Table 5.3, depending on the number of laptops in stock at the end of the previous day. Is the new policy better?

3 John has a standard die in his pocket (ie a cube with its six faces labelled from 1 to 6).

1. Describe how John can use the die to obtain realisations of the random variable \(X\), defined below.

   \(x\)	1	2	3
   \(\operatorname { Probability } ( X = x )\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 3 }\)

2. Describe how John can use the die to obtain realisations of the random variable \(Y\), defined below.

   \(y\)	1	2	3
   \(\operatorname { Probability } ( Y = y )\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)

3. John attempts to use the die to obtain a realisation of a uniformly distributed 2-digit random number. He throws the die 20 times. Each time he records one less than the number showing. He then adds together his 20 recorded numbers. Criticise John's methodology.

2 Graham skis each year in an Italian resort which shares a ski area with a Swiss resort. He can buy an Italian lift pass, or an international lift pass which gives him access to Switzerland as well as to Italy. For his 6-day holiday the Italian pass costs \(€ 200\) and the international pass costs \(€ 250\). If he buys an Italian pass then he can still visit Switzerland by purchasing day supplements at \(€ 30\) per day. If the weather is good during his holiday, then Graham visits Switzerland three times. If the weather is moderate he goes twice. If poor he goes once. If the weather is windy then the lifts are closed, and he is not able to go at all. In his years of skiing at the resort he has had good weather on \(30 \%\) of his visits, moderate weather on \(40 \%\), poor weather on \(20 \%\) and windy weather on \(10 \%\) of his visits.

1. Draw a decision tree to help Graham decide whether to buy an Italian lift pass or an international lift pass. Give the action he should take to minimize the EMV of his costs. When he arrives at the resort, and before he buys his lift pass, he finds that he has internet access to a local weather forecast, and to records of the past performance of the forecast. The 6-day forecast is limited to "good"/"not good", and the records show the actual weather proportions following those forecasts. It also shows that \(60 \%\) of historical forecasts have been "good" and \(40 \%\) "not good".

   \backslashbox{Forecast}{Actual}	good	moderate	poor	windy	proportion of forecasts
   good	0.4	0.5	0.1	0.0	0.6
   not good	0.15	0.25	0.35	0.25	0.4

2. Draw a decision tree to help Graham decide the worth of consulting the forecast before buying his lift pass. Give the actions he should take to minimize the EMV of his costs.

1 Keith is wondering whether or not to insure the value of his house against destruction. His friend Georgia has told him that it is a waste of money. Georgia argues that the insurance company sets its premiums (how much it charges for insurance) to take account of the probability of destruction, plus an extra fee for its profit. Georgia argues that house-owners are, on average, simply paying fees to the insurance company. Keith's house is valued at \(\pounds 400000\). The annual premium for insuring its value against destruction is \(\pounds 100\). Past statistics show that the probability of destruction in any one year is 0.0002 .

1. Draw a decision tree to model Keith's decision and the possible outcomes.
2. Compute Keith's EMV and give the course of action which corresponds to that EMV.
3. What would be the insurance premium if there were no fee for the insurance company? For the remainder of the question the insurance premium is still \(\pounds 100\).
   Suppose that, instead of EMV, Keith uses the utility function utility \(= ( \text { money } ) ^ { 0.5 }\).
4. Compute Keith's utility and give his corresponding course of action. Keith suspects that it may be the case that he lives in an area in which the probability of destruction in a given year, \(p\), is not 0.0002 .
5. Draw a decision tree, using the EMV criterion, to model Keith's decision in terms of \(p\), the probability of destruction in the area in which Keith lives.
6. Find the value of \(p\) which would make it worthwhile for Keith to insure his house using the EMV criterion.
7. Explain why Keith may wish to insure even if \(p\) is less than the value which you found in part (vi). [1]
   1. A national Sunday newspaper runs a "You are the umpire" series, in which questions are posed about whether a batsman in cricket is given "out", and why, or "not out". One Sunday the readers were told that a ball had either hit the bat and then the pad, or had missed the bat and hit the pad; the umpire could not be sure which. The ball had then flown directly to a fielder, who had caught it. The LBW (leg before wicket) rule is complicated. The readers were told that this batsman should be given out (LBW) if the ball had not hit the bat. On the other hand, if the ball had hit the bat, then he should be given out (caught). Readers were asked what the decision should be. The answer given in the newspaper was that this batsman should be given not out because the umpire could not be sure that the batsman was out (LBW), and could not be sure that he was out (caught).

1 Martin is considering paying for a vaccination against a disease. If he catches the disease he would not be able to work and would lose \(\pounds 900\) in income because he would have to stay at home recovering. The vaccination costs \(\pounds 20\). The vaccination would reduce his risk of catching the disease during the year from 0.02 to 0.001 .

1. Draw a decision tree for Martin.
2. Evaluate the EMV of Martin's loss at each node of your tree, and give the action that Martin should take to minimise the EMV of his loss. Martin can answer a medical questionnaire which will give an estimate of his susceptibility to the disease. If he is found to be susceptible, then his chance of catching the disease is 0.05 . Vaccination will reduce that to 0.0025 . If he is found not to be susceptible, then his chance of catching the disease is 0.01 and vaccination will reduce it to 0.0005 . Historically, \(25 \%\) of people are found to be susceptible.
3. What is the EMV of this questionnaire? Martin decides not to answer the questionnaire. He also decides that there is more than just his EMV to be considered in deciding whether or not to have the vaccination. The vaccination itself is likely to have side effects, but catching the disease would be very unpleasant. Martin estimates that he would find the effects of the disease 1000 times more unpleasant than the effects of the vaccination.
4. Analyse which course of action would minimise the unpleasantness for Martin.

1. Julia selects 3 letters at random, one at a time without replacement, from the word

\section*{VARIANCE} The discrete random variable \(X\) represents the number of times she selects a letter A.

1. Find the complete probability distribution of \(X\). Yuki selects 10 letters at random, one at a time with replacement, from the word \section*{DEVIATION}
2. Find the probability that he selects the letter E at least 4 times.

1. A biased 4 -sided spinner has the numbers \(6,7,8\) and 10 on it.

The discrete random variable \(X\) represents the score when the spinner is spun once and has the following probability distribution, where \(q\) is a probability.

1. Find the value of \(q\) Karen spins the spinner repeatedly until she either gets a 7 or she has taken 4 spins.
2. Show that the probability that Karen stops after taking her 3rd spin is 0.128 The random variable \(S\) represents the number of spins Karen takes.
3. Find the probability distribution for \(S\) The random variable \(N\) represents the number of times Karen gets a 7
4. Find \(\mathrm { P } ( S > N )\)

1. Two bags, \(\mathbf { A }\) and \(\mathbf { B }\), each contain balls which are either red or yellow or green.

Bag A contains 4 red, 3 yellow and \(n\) green balls.
Bag \(\mathbf { B }\) contains 5 red, 3 yellow and 1 green ball.
A ball is selected at random from bag \(\mathbf { A }\) and placed into bag \(\mathbf { B }\).
A ball is then selected at random from bag \(\mathbf { B }\) and placed into bag \(\mathbf { A }\).
The probability that bag \(\mathbf { A }\) now contains an equal number of red, yellow and green balls is \(p\). Given that \(p > 0\), find the possible values of \(n\) and \(p\).

5 A company needs to appoint 3 new assistants. 8 candidates are invited for interview; each candidate has a different surname. The candidates are to be interviewed one after another. The personnel officer randomly selects the order in which the candidates are to be interviewed by drawing their names out of a hat. One of the candidates is called Mr Browne and another is called Mrs Green.

1. Calculate the probability that Mr Browne is interviewed first and Mrs Green is interviewed last. 5 of the 8 candidates invited for interview are women and the other 3 are men. The chief executive can't make up his mind who to appoint so he randomly selects 3 candidates by drawing their names out of a hat.
2. Determine the probability that more women than men are selected.

5

1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls.
   Find the probability that the team contains no more than 3 girls.
2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).

4 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other.
2. Find the probability that no two men are next to one another.

8 Alex claims that he can read people's minds. A volunteer, Jane, arranges the integers 1 to \(n\) in an order of Jane's own choice and Alex tells Jane what order he believes was chosen. They agree that Alex's claim will be accepted if he gets the order completely correct or if he gets the order correct apart from two numbers which are the wrong way round. They use a value of \(n\) such that, if Alex chooses the order of the integers at random, the probability that Alex's claim will be accepted is less than \(1 \%\). Determine the smallest possible value of \(n\). \section*{END OF QUESTION PAPER}

6 A teacher has 10 different mathematics books. Of these books, 5 are on Algebra, 3 are on Calculus and 2 are on Trigonometry. The teacher chooses 5 of the books at random.

1. Find the probability that 3 of the books are on Algebra. The teacher now arranges all 10 books in random order on a shelf.
2. Find the probability that the Calculus books are next to each other and the Trigonometry books are next to each other. \section*{In this question you must show detailed reasoning.}
3. Find the probability that 2 of the Calculus books are next to each other but the third Calculus book is separated from the other 2 by at least 1 other book.

2 A music lover has 30 CDs arranged in a random order in a line on a shelf. Of these CDs, 7 are classed as Baroque, 10 as Classical and 13 as Romantic.

1. Determine the probability that all 7 Baroque CDs are next to each other.
2. Determine the probability that, of the 10 CDs furthest to the left on the shelf, at least 6 are Baroque.

7 A bag contains \(2 m\) yellow and \(m\) green counters. Three counters are chosen at random, without replacement. The probability that exactly two of the three counters are yellow is \(\frac { 28 } { 55 }\). Determine the value of \(m\).

7 The 20 members of a club consist of 10 Town members and 10 Country members.

1. All 20 members are arranged randomly in a straight line. Determine the probability that the Town members and the Country members alternate.
2. Ten members of the club are chosen at random. Show that the probability that the number of Town members chosen is no more than \(r\), where \(0 \leqslant r \leqslant 10\), is given by \(\frac { 1 } { \mathrm {~N} } \sum _ { \mathrm { i } = 0 } ^ { \mathrm { r } } \left( { } ^ { 10 } \mathrm { C } _ { \mathrm { i } } \right) ^ { 2 }\) where \(N\) is an integer to be determined.

2 Mo eats exactly 6 doughnuts in 4 days.

1. What does the pigeonhole principle tell you about the number of doughnuts Mo eats in a day? Mo eats exactly 6 doughnuts in 4 days, eating at least 1 doughnut each day.
2. Show that there must be either two consecutive days or three consecutive days on which Mo eats a total of exactly 4 doughnuts. Mo eats exactly 3 identical jam doughnuts and exactly 3 identical iced doughnuts over the 4 days.
   The number of jam doughnuts eaten on the four days is recorded as a list, for example \(1,0,2,0\). The number of iced doughnuts eaten is not recorded.
3. Show that 20 different such lists are possible.

1 Alfie has a set of 15 cards numbered consecutively from 1 to 15.
He chooses two of the cards.

1. How many different sets of two cards are possible? Alfie places the two cards side by side to form a number with 2,3 or 4 digits.
2. Explain why there are fewer than \({ } ^ { 15 } \mathrm { P } _ { 2 } = 210\) possible numbers that can be made.
3. Explain why, with these cards, 1 is the lead digit more often than any other digit. Alfie makes the number 113, which is a 3-digit prime number. Alfie says that the problem of working out how many 3-digit prime numbers can be made using two of the cards is a construction problem, because he is trying to find all of them.
4. Explain why Alfie is wrong to say this is a construction problem.

3 Heidi has a pack of cards.
Each card has a single digit on one side and is blank on the other side.
Each of the digits from 1 to 9 appears on exactly four cards.
Apart from the numerical values of the digits, the cards are indistinguishable from each other.
Heidi draws four cards from the pack, at random and without replacement. She places the four cards in a row to make a four-digit number. Determine how many different four-digit numbers Heidi could have made in each of the following cases.

1. The four digits are all different.
2. Two of the digits are the same and the other two digits are different.
3. There is no restriction on whether any of the digits are the same or not.

2 Some of the activities that may be involved in making a cup of tea are listed below. A: Boil water.
B: Put teabag in teapot, pour on boiled water and let tea brew.
C: Get cup from cupboard.
D: Pour tea into cup.
E: Add milk to cup.
F: Add sugar to cup. Activity A must happen before activity B.
Activities B and C must happen before activity D .
Activities E and F cannot happen until after activity C.
Other than that, the activities can happen in any order.

1. Lisa does not take milk or sugar in her tea, so she only needs to use activities \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . In how many different orders can activities \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D be arranged, subject to the restrictions above?
2. Mick takes milk but no sugar, so he needs to use activities \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E . Explain carefully why there are exactly nine different orders for these activities, subject to the restrictions above.
3. Find the number of different orders for all six activities, subject to the restrictions above. Explain your reasoning carefully.

3 Six red counters and four blue counters are arranged in a straight line in a random order.
Find the probability that

1. no blue counter has fewer than two red counters between it and the nearest other blue counter,
2. no two blue counters are next to one another.