5.01a Permutations and combinations: evaluate probabilities

The students in a large Sixth Form can choose to do exactly one of Community Service, Games or Private Study on Wednesday afternoons. The probabilities that a randomly chosen student does Games and Private Study are \(\frac{3}{8}\) and \(\frac{1}{5}\) respectively. It may be assumed that the number of students is large enough for these probabilities to be treated as constant.

1. Find the probability that a randomly chosen student does Community Service. [2 marks]
2. If two students are chosen at random, find the probability that they both do the same activity. [3 marks]
3. If three students are chosen at random, find the probability that exactly one of them does Games. [3 marks]

Two-fifths of the students are girls, and a quarter of these girls do Private Study.

1. Find the probability that a randomly chosen student who does Private Study is a boy. [5 marks]

A washing-up bowl contains 6 spoons, 5 forks and 3 knives. Three of these 14 items are removed at random, without replacement. Find the probability that

1. all three items are of different kinds, [3]
2. all three items are of the same kind. [3]

The five letters of the word NEVER are arranged in random order in a straight line.

1. How many different orders of the letters are possible? [2]
2. In how many of the possible orders are the two Es next to each other? [2]
3. Find the probability that the first two letters in the order include exactly one letter E. [3]

Three letters are selected at random from the 8 letters of the word COMPUTER, without regard to order.

1. Find the number of possible selections of 3 letters. [2]
2. Find the probability that the letter P is included in the selection. [3]

Three letters are now selected at random, one at a time, from the 8 letters of the word COMPUTER, and are placed in order in a line.

1. Find the probability that the 3 letters form the word TOP. [3]

The menu below shows all the dishes available at a certain restaurant. A group of friends decide that they will share a total of 2 different rice dishes, 3 different main dishes and 4 different vegetable dishes from this menu. Given these restrictions,

1. find the number of possible combinations of dishes that they can choose to share, [3]
2. assuming that all choices are equally likely, find the probability that they choose boiled rice. [2]

The friends decide to add a further restriction as follows. If they choose boiled rice, they will not choose potatoes.

1. Find the number of possible combinations of dishes that they can now choose. [3]

The diagram shows five cards, each with a letter on it. \includegraphics{figure_6} The letters A and E are vowels; the letters B, C and D are consonants.

1. Two of the five cards are chosen at random, without replacement. Find the probability that they both have vowels on them. [2]
2. The two cards are replaced. Now three of the five cards are chosen at random, without replacement. Find the probability that they include exactly one card with a vowel on it. [3]
3. The three cards are replaced. Now four of the five cards are chosen at random without replacement. Find the probability that they include the card with the letter B on it. [2]

My credit card has a 4-digit code called a PIN. You should assume that any 4-digit number from 0000 to 9999 can be a PIN.

1. If I cannot remember any digits and guess my number, find the probability that I guess it correctly. [1]

In fact my PIN consists of four different digits. I can remember all four digits, but cannot remember the correct order.

1. If I now guess my number, find the probability that I guess it correctly. [2]

Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if

1. no student may win more than 1 prize, [2]
2. no student may win all 3 prizes. [2]

There are 13 men and 10 women in a running club. A committee of 3 men and 3 women is to be selected.

1. In how many different ways can the three men be selected? [2]
2. In how many different ways can the whole committee be selected? [2]
3. A random sample of 6 people is selected from the running club. Find the probability that this sample consists of 3 men and 3 women. [2]

I have 5 books, each by a different author. The authors are Austen, Brontë, Clarke, Dickens and Eliot.

1. If I arrange the books in a random order on my bookshelf, find the probability that the authors are in alphabetical order with Austen on the left. [2]
2. If I choose two of the books at random, find the probability that I choose the books written by Austen and Brontë. [3]

The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

Heating quality (\(x\))	9.1 \(\leqslant x <\) 9.3	9.3 \(< x \leqslant\) 9.5	9.5 \(< x \leqslant\) 9.7	9.7 \(< x \leqslant\) 9.9	9.9 \(< x \leqslant\) 10.1
Frequency	5	7	15	16	7

1. Draw a cumulative frequency diagram to illustrate these data. [5]
2. Use the diagram to estimate the median and interquartile range of the data. [3]
3. Show that there are no outliers in the sample. [3]
4. Three of these 50 sacks are selected at random. Find the probability that
   1. in all three, the heating quality \(x\) is more than 9.5, [3]
   2. in at least two, the heating quality \(x\) is more than 9.5. [4]

There are 16 girls and 14 boys in a class. Four of them are to be selected to form a quiz team. The team is to be selected at random.

1. Find the probability that all 4 members of the team will be girls. [3]
2. Find the probability that the team will contain at least one girl and at least one boy. [3]

A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a £10 prize, 20 of them have a £100 prize, one of them has a £5000 prize and all of the rest have no prize. This information is summarised in the frequency table below.

Prize money	£0	£10	£100	£5000
Frequency	9929	50	20	1

1. Find the mean and standard deviation of the prize money per ticket. [4]
2. I buy two of these tickets at random. Find the probability that I win either two £10 prizes or two £100 prizes. [3]

A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$F(t) = \begin{cases} 0 & t \leq 0 \\ 1 - e^{-\frac{1}{t}} & t > 0 \end{cases}$$

1. Find the median delay and the 90th percentile delay. [5]
2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay. [5]
3. Find the probability that a delay lasts longer than the mean delay. [2]

You are given that the variance of \(T\) is 9.

1. Let \(\overline{T}\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\overline{T}\). [3]
2. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling? [3]

Find the value of \(\frac{100!}{98! \times 3!}\) Circle your answer. [1 mark] \(\frac{50}{147}\) \quad \(1650\) \quad \(3300\) \quad \(161700\)

26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.

1. Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels. [4]
2. All 26 cards are arranged in a line, in random order.
   1. Show that the probability that all the vowels are next to one another is \(\frac{1}{2990}\). [3]
   2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [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. [3]
2. Find the probability that no two men are next to one another. [4]

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 \(n\). [7]

The letters of the word CHAFFINCH are written on cards.

1. In how many ways can the letters be rearranged with no restrictions. [1]
2. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them. [3]

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. [3]
2. Find the probability that no two men are next to one another. [4]

The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it. \includegraphics{figure_6} 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? [3]
2. Find the probability that the seven cards include exactly one card showing the letter A. [4]