5.01b Selection/arrangement: probability problems

6 A group of 7 students sit in random order on a bench.

1. (a) Find the number of orders in which they can sit.
   (b) The 7 students include Tom and Jerry. Find the probability that Tom and Jerry sit next to each other.
2. The students consist of 3 girls and 4 boys. Find the probability that
   (a) no two boys sit next to each other,
   (b) all three girls sit next to each other.

7

1. 5 of the 7 letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) are arranged in a random order in a straight line.
   1. How many different arrangements of 5 letters are possible?
   2. How many of these arrangements end with a vowel (A or E)?
   3. A group of 5 people is to be chosen from a list of 7 people.
      (a) How many different groups of 5 people can be chosen?
      (b) The list of 7 people includes Jill and Jo. A group of 5 people is chosen at random from the list. Given that either Jill and Jo are both chosen or neither of them is chosen, find the probability that both of them are chosen.

8 A group of 8 people, including Kathy, David and Harpreet, are planning a theatre trip.

1. Four of the group are chosen at random, without regard to order, to carry the refreshments. Find the probability that these 4 people include Kathy and David but not Harpreet.
2. The 8 people sit in a row. Kathy and David sit next to each other and Harpreet sits at the left-hand end of the row. How many different arrangements of the 8 people are possible?
3. The 8 people stand in a line to queue for the exit. Kathy and David stand next to each other and Harpreet stands next to them. How many different arrangements of the 8 people are possible?

6

1. The seven digits \(1,1,2,3,4,5,6\) are arranged in a random order in a line. Find the probability that they form the number 1452163.
2. Three of the seven digits \(1,1,2,3,4,5,6\) are chosen at random, without regard to order.
   1. How many possible groups of three digits contain two 1s?
   2. How many possible groups of three digits contain exactly one 1?
   3. How many possible groups of three digits can be formed altogether?

4 At a dog show, three out of eleven dogs are to be selected for a national competition.

1. Find the number of possible selections.
2. Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.

2 There are 14 girls and 11 boys in a class. A quiz team of 5 students is to be chosen from the class.

1. How many different teams are possible?
2. If the team must include 3 girls and 2 boys, find how many different teams are possible.

2 Thomas has six tiles, each with a different letter of his name on it.

1. Thomas arranges these letters in a random order. Find the probability that he arranges them in the correct order to spell his name.
2. On another occasion, Thomas picks three of the six letters at random. Find the probability that he picks the letters T, O and M (in any order).

3

1. There are 5 runners in a race. How many different finishing orders are possible? [You should assume that there are no 'dead heats', where two runners are given the same position.] For the remainder of this question you should assume that all finishing orders are equally likely.
2. The runners are denoted by \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\). Find the probability that they either finish in the order ABCDE or in the order EDCBA.
3. Find the probability that the first 3 runners to finish are \(\mathrm { A } , \mathrm { B }\) and C , in that order.
4. Find the probability that the first 3 runners to finish are \(\mathrm { A } , \mathrm { B }\) and C , in any order.

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]
   (a) 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 \(\mathrm { P } ( S > N )\)

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.

6 A bag contains 6 identical blue counters and 5 identical yellow counters.

1. Three counters are selected at random, without replacement. Find the probability that at least two of the counters are blue. All 11 counters are now arranged in a row in a random order.
2. Find the probability that all the yellow counters are next to each other.
3. Find the probability that no yellow counter is next to another yellow counter.
4. Find the probability that the counters are arranged in such a way that both of the following conditions hold.

2 A project is represented by the activity network and cascade chart below. The table showing the number of workers needed for each activity is incomplete. Each activity needs at least 1 worker. \includegraphics[max width=\textwidth, alt={}, center]{7717b4ca-45ab-4111-9f59-5a3abb04b388-2_202_565_1605_201} \includegraphics[max width=\textwidth, alt={}, center]{7717b4ca-45ab-4111-9f59-5a3abb04b388-2_328_560_1548_820}

1. Complete the table in the Printed Answer Booklet to show the immediate predecessors for each activity.
2. Calculate the latest start time for each non-critical activity. The minimum number of workers needed is 5 .
3. What type of problem (existence, construction, enumeration or optimisation) is the allocation of a number of workers to the activities? There are 8 workers available who can do activities A and B .
4. 1. Find the number of ways that the workers for activity A can be chosen.
   2. When the workers have been chosen for activity A , find the total number of ways of choosing the workers for activity B for all the different possible values of x , where \(\mathrm { x } \geqslant 1\).

3 A para relay team of 4 swimmers needs to be chosen from a group of 7 swimmers.

1. How many ways are there to choose 4 swimmers from 7? There are no restrictions on how many men and how many women are in the team. The group of 7 swimmers consists of 5 men and 2 women.
2. How many ways are there to choose a team with more men than women? The physical impairment of each swimmer is given a score.
   The scores for the swimmers are \(\begin{array} { l l l l l l l } 3 & 4 & 4 & 6 & 7 & 8 & 9 \end{array}\) The total score for the team must be 20 or less.
3. How many different valid teams are possible? The order of the swimmers in the team is now taken into consideration.
4. In total, how many different arrangements are there of valid teams?
5. In how many of these valid teams are the scores of the swimmers in increasing order? For example, 3, 4, 4, 8 but not 4, 3, 4, 8 .

1. A small sports club has 12 adult members and 14 junior members.

The club needs to enter a team of 8 players for a particular competition.
Determine the number of ways in which the team can be selected if

1. there are no restrictions on the team,
2. the team must contain 4 adults and 4 juniors,
3. more than half the team must be adults.

6 Answer parts (ii)(A) and (iii)(B) of this question on the insert provided. A particular component of a machine sometimes fails. The probability of failure depends on the age of the component, as shown in Table 6. \section*{Table 6} You are to simulate six years of machine operation to estimate the probability of the component failing during that time. This will involve you using six 2-digit random numbers, one for each year.

1. Give a rule for using a 2-digit random number to simulate failure of the component in its first year of life. Similarly give rules for simulating failure during each of years 2 to 6 .
2. (A) Use your rules, together with the random numbers given in the insert, to complete the simulation table in the insert. This simulates 10 repetitions of six years operation of the machine. Start in the first column working down cell-by-cell. In each cell enter a tick if there is no simulated failure and a cross if there is a simulated failure. Stop and move on to the next column if a failure occurs.
   (B) Use your results to estimate the probability of a failure occurring. It is suggested that any component that has not failed during the first three years of its life should automatically be replaced.
3. (A) Describe how to simulate the operation of this policy.
   (B) Use the table in the insert to simulate 10 repetitions of the application of this policy. Re-use the same random numbers that are given in the insert.
   (C) Use your results to estimate the probability of a failure occurring.
4. How might the reliability of your estimates in parts (ii) and (iii) be improved?

2 Each letter of the words NEW COURSE is written on a card (including one blank card, representing the space between the words), so that there are 10 cards altogether.

1. All 10 cards are arranged in a random order in a straight line. Find the probability that the two cards containing an E are next to each other.
2. 4 cards are chosen at random. Find the probability that at least three consonants ( \(\mathrm { N } , \mathrm { W } , \mathrm { C } , \mathrm { R } , \mathrm { S }\) ) are on the cards chosen.

6 A bag contains 7 red counters and 5 blue counters.

1. Fred chooses 4 counters at random, without replacement. Show that the probability that Fred chooses exactly 2 red counters is \(\frac { 14 } { 33 }\).
2. Lina chooses 4 counters at random from the bag, records whether or not exactly 2 red counters are chosen, and returns the counters to the bag. She carries out this experiment 99 times.
   1. Find the mean of the number of experiments that result in choosing exactly 2 red counters.
   2. Find the variance of the number of experiments that result in choosing exactly 2 red counters.
   3. Alex arranges all 12 counters in a random order in a straight line. A is the event: no two blue counters are next to one another. B is the event: all the blue counters are next to one another. Find \(\mathrm { P } ( A \cup B )\).

3

1. Alex places 20 black counters and 8 white counters into a bag. She removes 8 counters at random without replacement. Find the probability that the bag now contains exactly 5 white counters.
2. Bill arranges 8 blue counters and 4 green counters in a random order in a straight line. Find the probability that exactly three of the green counters are next to one another.

7 A committee of 7 people is to be chosen at random from 18 volunteers.

1. In how many different ways can the committee be chosen? The 18 volunteers consist of 5 people from Gloucester, 6 from Hereford and 7 from Worcester. The committee is to be chosen randomly. Find the probability that the committee will
2. consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester,
3. include exactly 5 people from Worcester,
4. include at least 2 people from each of the three cities. 1 Jenny and John are each allowed two attempts to pass an examination.
5. Jenny estimates that her chances of success are as follows.
   • The probability that she will pass on her first attempt is \(\frac { 2 } { 3 }\).
   • If she fails on her first attempt, the probability that she will pass on her second attempt is \(\frac { 3 } { 4 }\). Calculate the probability that Jenny will pass.
   • John estimates that his chances of success are as follows.
   • The probability that he will pass on his first attempt is \(\frac { 2 } { 3 }\).
   • Overall, the probability that he will pass is \(\frac { 5 } { 6 }\).
   Calculate the probability that if John fails on his first attempt, he will pass on his second attempt. 2 For each of 50 plants, the height, \(h \mathrm {~cm}\), was measured and the value of ( \(h - 100\) ) was recorded. The mean and standard deviation of \(( h - 100 )\) were found to be 24.5 and 4.8 respectively.
6. Write down the mean and standard deviation of \(h\). The mean and standard deviation of the heights of another 100 plants were found to be 123.0 cm and 5.1 cm respectively.
7. Describe briefly how the heights of the second group of plants compare with the first.
8. Calculate the mean height of all 150 plants. 3 In Mr Kendall's cupboard there are 3 tins of baked beans and 2 tins of pineapple. Unfortunately his daughter has removed all the labels for a school project and so the tins are identical in appearance. Mr Kendall wishes to use both tins of pineapple for a fruit salad. He opens tins at random until he has opened the two tins of pineapples. Let \(X\) be the number of tins that Mr Kendall opens.
9. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 5 }\).
10. The probability distribution of \(X\) is given in the table below.

    \(x\)	2	3	4	5
    \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 10 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 2 } { 5 }\)

    Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

8 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are positive constants.

1. Find \(k\) in terms of \(n\).
2. Show that \(\mathrm { E } ( X ) = \frac { n + 1 } { n + 2 }\). It is given that \(n = 3\).
3. Find the variance of \(X\).
4. One hundred observations of \(X\) are taken, and the mean of the observations is denoted by \(\bar { X }\). Write down the approximate distribution of \(\bar { X }\), giving the values of any parameters.
5. Write down the mean and the variance of the random variable \(Y\) with probability density function given by $$g ( y ) = \begin{cases} 4 \left( y + \frac { 4 } { 5 } \right) ^ { 3 } & - \frac { 4 } { 5 } \leqslant y \leqslant \frac { 1 } { 5 } \\ 0 & \text { otherwise } \end{cases}$$

2 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.