5.01a Permutations and combinations: evaluate probabilities

1. A bag contains 19 red beads and 1 blue bead only.

Linda selects a bead at random from the bag. She notes its colour and replaces the bead in the bag. She then selects a second bead at random from the bag and notes its colour. Find the probability that

1. both beads selected are blue,
2. exactly one bead selected is red. In another bag there are 9 beads, 4 of which are green and the rest are yellow.
   Linda selects 3 beads from this bag at random without replacement.
3. Find the probability that 2 of these beads are yellow and 1 is green. Linda replaces the 3 beads and then selects another 4 at random without replacement.
4. Find the probability that at least 1 of the beads is green.

6 A group of 8 friends travels to the airport in two taxis, \(P\) and \(Q\). Each taxi can take 4 passengers.

1. The 8 friends divide themselves into two groups of 4, one group for taxi \(P\) and one group for taxi \(Q\), with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_272_456_242_461} \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_281_455_233_1151} Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi \(P\) and Jon and Sarah sit in the back of taxi \(P\) next to each other.
2. Find the number of different seating arrangements that are now possible for the 8 friends.
   

6 A test consists of 4 algebra questions, A, B, C and D, and 4 geometry questions, G, H, I and J.
The examiner plans to arrange all 8 questions in a random order, regardless of topic.

1. (a) How many different arrangements are possible?
   (b) Find the probability that no two Algebra questions are next to each other and no two Geometry questions are next to each other. Later, the examiner decides that the questions should be arranged in two sections, Algebra followed by Geometry, with the questions in each section arranged in a random order.
2. (a) How many different arrangements are possible?
   (b) Find the probability that questions A and H are next to each other.
   (c) Find the probability that questions B and J are separated by more than four other questions.

6

1. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number.

   1

   3

   3

   3

   5

   5

   9

   How many different 7 -digit numbers can be formed using these cards?
2. The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it. \includegraphics[max width=\textwidth, alt={}, center]{98ac515d-fd47-4864-afd6-321e9848d6cb-04_398_801_596_632} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
   1. How many selections of seven cards are possible?
   2. Find the probability that the seven cards include exactly one card showing the letter A .

9 A bag contains 9 discs numbered 1, 2, 3, 4, 5, 6, 7, 8, 9 .

1. Andrea chooses 4 discs at random, without replacement, and places them in a row.
   1. How many different 4 -digit numbers can be made?
   2. How many different odd 4-digit numbers can be made?
   3. Andrea's 4 discs are put back in the bag. Martin then chooses 4 discs at random, without replacement. Find the probability that
      (a) the 4 digits include at least 3 odd digits,
      (b) the 4 digits add up to 28 .

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.

4 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.

1. Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
2. Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
3. Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.

1 At a garden centre there is a box containing 50 hyacinth bulbs. Of these, 30 will produce a blue flower and the remaining 20 will produce a red flower. Unfortunately they have become mixed together so that it is not known which of the bulbs will produce a blue flower and which will produce a red flower. Karen buys 3 of these bulbs.

1. Find the probability that all 3 of these bulbs will produce blue flowers.
2. Find the probability that Karen will have at least one flower of each colour from her 3 bulbs.

2 An examination paper consists of two sections. Section A has 5 questions and Section B has 9 questions. Candidates are required to answer 6 questions.

1. In how many different ways can a candidate choose 6 questions, if 3 are from Section A and 3 are from Section B?
2. Another candidate randomly chooses 6 questions to answer. Find the probability that this candidate chooses 3 questions from each section.

2 Every evening, 5 men and 5 women are chosen to take part in a phone-in competition. Of these 10 people, exactly 3 will win a prize. These 3 prize-winners are chosen at random.

1. Find the probability that, on a particular evening, 2 of the prize-winners are women and the other is a man. Give your answer as a fraction in its simplest form.
2. Four evenings are selected at random. Find the probability that, on at least three of the four evenings, 2 of the prize-winners are women and the other is a man.

4 A rugby team of 15 people is to be selected from a squad of 25 players.

1. How many different teams are possible?
2. In fact the team has to consist of 8 forwards and 7 backs. If 13 of the squad are forwards and the other 12 are backs, how many different teams are now possible?
3. Find the probability that, if the team is selected at random from the squad of 25 players, it contains the correct numbers of forwards and backs.

8 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box. \includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-4_287_1177_319_427}

1. Find the interquartile range of the data and hence determine whether there are any outliers at either end of the distribution. Ben buys a box of these chocolates each weekend. The chocolates all look the same on the outside, but 7 of them have orange centres, 6 have cherry centres, 4 have coffee centres and 3 have lemon centres. One weekend, each of Ben's 3 children eats one of the chocolates, chosen at random.
2. Calculate the probabilities of the following events. A: all 3 chocolates have orange centres \(B\) : all 3 chocolates have the same centres
3. Find \(\mathrm { P } ( A \mid B )\) and \(\mathrm { P } ( B \mid A )\). The following weekend, Ben buys an identical box of chocolates and again each of his 3 children eats one of the chocolates, chosen at random.
4. Find the probability that, on both weekends, the 3 chocolates that they eat all have orange centres.
5. Ben likes all of the chocolates except those with cherry centres. On another weekend he is the first of his family to eat some of the chocolates. Find the probability that he has to select more than 2 chocolates before he finds one that he likes. \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

4 A timber supplier cuts wooden fence posts from felled trees. The posts are of length \(( k + X ) \mathrm { cm }\) where \(k\) is a constant and \(X\) is a random variable which has probability density function $$f ( x ) = \begin{cases} 1 + x & - 1 \leqslant x < 0 \\ 1 - x & 0 \leqslant x \leqslant 1 \\ 0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Write down the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
3. Write down, in terms of \(k\), the approximate distribution of \(\bar { L }\), the mean length of a random sample of 50 fence posts. Justify your choice of distribution.
4. In a particular sample of 50 posts, the mean length is 90.06 cm . Find a \(95 \%\) confidence interval for the true mean length of the fence posts.
5. Explain whether it is reasonable to suppose that \(k = 90\).

4 At the school summer fair, one of the games involves throwing darts at a circular dartboard of radius \(a\) lying on the ground some distance away. Only darts that land on the board are counted. The distance from the centre of the board to the point where a dart lands is modelled by the random variable \(R\). It is assumed that the probability that a dart lands inside a circle of radius \(r\) is proportional to the area of the circle.

1. By considering \(\mathrm { P } ( R < r )\) show that \(\mathrm { F } ( r )\), the cumulative distribution function of \(R\), is given by $$\mathrm { F } ( r ) = \begin{cases} 0 & r < 0 , \\ \frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leqslant r \leqslant a , \\ 1 & r > a . \end{cases}$$
2. Find \(\mathrm { f } ( r )\), the probability density function of \(R\).
3. Find \(\mathrm { E } ( R )\) and show that \(\operatorname { Var } ( R ) = \frac { a ^ { 2 } } { 18 }\). The radius \(a\) of the dartboard is 22.5 cm .
4. Let \(\bar { R }\) denote the mean distance from the centre of the board of a random sample of 100 darts. Write down an approximation to the distribution of \(\bar { R }\).
5. A random sample of 100 darts is found to give a mean distance of 13.87 cm . Does this cast any doubt on the modelling?

2

1. 1. Give two reasons why an investigator might need to take a sample in order to obtain information about a population.
   2. State two requirements of a sample.
   3. Discuss briefly the advantage of the sampling being random.
2. 1. Under what circumstances might one use a Wilcoxon single sample test in order to test a hypothesis about the median of a population? What distributional assumption is needed for the test?
   2. On a stretch of road leading out of the centre of a town, highways officials have been monitoring the speed of the traffic in case it has increased. Previously it was known that the median speed on this stretch was 28.7 miles per hour. For a random sample of 12 vehicles on the stretch, the following speeds were recorded. $$\begin{array} { l l l l l l l l l l l l } 32.0 & 29.1 & 26.1 & 35.2 & 34.4 & 28.6 & 32.3 & 28.5 & 27.0 & 33.3 & 28.2 & 31.9 \end{array}$$ Carry out a test, with a \(5 \%\) significance level, to see whether the speed of the traffic on this stretch of road seems to have increased on the whole.
      [0pt] [10]

Aram is a packer at a supermarket checkout and the time he takes to pack a randomly chosen item has mean 1.5 s and standard deviation 0.4 s . Justifying any approximation that you make, find the probability that Aram will pack 50 randomly chosen items in less than 70 s . Find the greatest number of items that Aram could pack within 70 s with probability at least \(90 \%\). Huldu is also a packer at the supermarket. The time that she takes to pack a randomly chosen item has mean 1.3 s and standard deviation 0.5 s . Aram and Huldu each have 50 items to pack. Find the probability that Huldu takes a shorter time than Aram.

10 Jill and Kate are playing a game as a practice for a penalty shoot-out. They take alternate turns at kicking a football at a goal. The probability that Jill will score a goal with any kick is \(\frac { 1 } { 3 }\), independently of previous outcomes. The probability that Kate will score a goal with any kick is \(\frac { 1 } { 4 }\), independently of previous outcomes. Jill begins the game. If Jill is the first to score, then Kate is allowed one more kick. If Kate scores with this kick, then the game is a draw, but if she does not score then Jill wins the game. If Kate is the first to score, then she wins the game, and no further kicks are taken.

1. Show that the probability that Jill scores on her 5th kick is \(\frac { 1 } { 48 }\).
2. Show that the probability that Kate wins the game on her \(n\)th kick is \(\frac { 1 } { 3 \times 2 ^ { n } }\).
3. Find the probability that Jill wins the game.
4. Find the probability that the game is a draw.

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.

5 There is an insert for use in parts (iii) and (iv) of this question.
This question concerns the simulation of cars passing through two sets of pedestrian controlled traffic lights. The time intervals between cars arriving at the first set of lights are distributed according to Table 5.1. \begin{table}[h] \end{table}

1. Give an efficient rule for using two-digit random numbers to simulate arrival intervals.
2. Use two-digit random numbers from the list below to simulate the arrival times of five cars at the first lights. The first car arrives at the time given by the first arrival interval. Random numbers: \(24,01,99,89,77,19,58,42\) The two sets of traffic lights are 23 seconds driving time apart. Moving cars are always at least 2 seconds apart. If there is a queue at a set of lights, then when the red light ends the first car in the queue moves off immediately, the second car 2 seconds later, the third 2 seconds after that, etc. In this simple model there is to be no consideration of accelerations or decelerations, and the lights are either red or green. Table 5.2 shows the times when the lights are red. \begin{table}[h]

   \multirow{2}{*}{ first set of lights }	red start time	14	50	105	155
   \cline { 2 - 6 }	red end time	29	65	120	170
   \multirow{2}{*}{ second set of lights }	red start time	10	55	105	150
   \cline { 2 - 6 }	red end time	25	70	120	165

   \captionsetup{labelformat=empty} \caption{Table 5.2}
   \end{table}
3. Complete the table in the insert to simulate the passage of 10 cars through both sets of traffic lights. Use the arrival times given there.
4. Find the mean delay experienced by these cars in passing through each set of lights.
5. How could the output from this simulation model be made more reliable?