2.03e Model with probability: critiquing assumptions

1

1. How many different arrangements are there of the 11 letters in the word MISSISSIPPI?
2. Two letters are chosen at random from the 11 letters in the word MISSISSIPPI. Find the probability that these two letters are the same.

3 The probability that a certain spinner lands on red on any spin is \(p\). The spinner is spun 140 times and it lands on red 35 times.

1. Find an approximate \(96 \%\) confidence interval for \(p\).
   From three further experiments, Jack finds a 90\% confidence interval, a 95\% confidence interval and a 99\% confidence interval for \(p\).
2. Find the probability that exactly two of these confidence intervals contain the true value of \(p\).

1. A disc of radius 1 cm is rolled onto a horizontal grid of rectangles so that the disc is equally likely to land anywhere on the grid. Each rectangle is 5 cm long and 3 cm wide. There are no gaps between the rectangles and the grid is sufficiently large so that no discs roll off the grid.

If the disc lands inside a rectangle without covering any part of the edges of the rectangle then a prize is won. By considering the possible positions for the centre of the disc,

1. show that the probability of winning a prize on any particular roll is \(\frac { 1 } { 5 }\) A group of 15 students each roll the disc onto the grid twenty times and record the number of times, \(x\), that each student wins a prize. Their results are summarised as follows $$\sum x = 61 \quad \sum x ^ { 2 } = 295$$
2. Find the standard deviation of the number of prizes won per student. A second group of 12 students each roll the disc onto the grid twenty times and the mean number of prizes won per student is 3.5 with a standard deviation of 2
3. Find the mean and standard deviation of the number of prizes won per student for the whole group of 27 students. The 27 students also recorded the number of times that the disc covered a corner of a rectangle and estimated the probability to be 0.2216 (to 4 decimal places).
4. Explain how this probability could be used to find an estimate for the value of \(\pi\) and state the value of your estimate.

6. The Venn diagram below shows the probabilities of customers having various combinations of a starter, main course or dessert at Polly's restaurant. \(S =\) the event a customer has a starter. \(M =\) the event a customer has a main course. \(D =\) the event a customer has a dessert. \includegraphics[max width=\textwidth, alt={}, center]{fa0dbe16-ace8-4c44-8404-2bc4e1879d57-10_602_1125_607_413} Given that the events \(S\) and \(D\) are statistically independent

1. find the value of \(p\).
2. Hence find the value of \(q\).
3. Find
   1. \(\quad\) P( \(D \mid M \cap S\) )
   2. \(\operatorname { P } \left( D \mid M \cap S ^ { \prime } \right)\) One evening 63 customers are booked into Polly's restaurant for an office party. Polly has asked for their starter and main course orders before they arrive. Of these 63 customers 27 ordered a main course and a starter, 36 ordered a main course without a starter.
4. Estimate the number of desserts that these 63 customers will have.

1. The Venn diagram shows three events, \(A\), \(B\) and \(C\), and their associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{e73193ee-339e-48ab-811c-9ab6817f786d-04_680_780_296_644}

Events \(B\) and \(C\) are mutually exclusive.
Events \(A\) and \(C\) are independent.
Showing your working, find the value of \(x\), the value of \(y\) and the value of \(z\).

3 An auction house offers items of jewellery for sale at its public auctions. Each item has a reserve price which is less than the lower price estimate which, in turn, is less than the upper price estimate. The outcome for any item is independent of the outcomes for all other items. The auction house has found, from past records, the following probabilities for the outcomes of items of jewellery offered for sale.

2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability \(p\) of hitting the target. Successive shots are independent.

1. Determine the probability that Jeff wins the game
   i) with his first shot,
   ii) with his second shot.
2. Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
3. Find the range of values of \(p\) for which Mary is more likely to win the game than Jeff.

A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.

1. Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work. [3]
2. Use the random numbers in your answer book to run your simulation 5 times, recording your results. [2]
3. From your results compute an estimate of the mean number of draws needed to get a vowel. [2]
4. State how you could produce a more accurate estimate. [1]

The scatter diagram uses information about all the Local Authorities (LAs) in the UK, taken from the 2011 census. For each LA it shows the percentage (\(x\)) of employees who used public transport to travel to work and the percentage (\(y\)) who used motorised private transport. "Public transport" includes train, bus, minibus, coach, underground, metro and light rail. "Motorised private transport" includes car, van, motorcycle, scooter, moped, taxi and passenger in a car or van. \includegraphics{figure_13}

1. Most of the points in the diagram lie on or near the line with equation \(x + y = k\), where \(k\) is a constant.
   1. Give a possible value for \(k\). [1]
   2. Hence give an approximate value for the percentage of employees who either worked from home or walked or cycled to work. [1]
2. The average amount of fuel used per person per day for travelling to work in any LA is denoted by F. Consider the two groups of LAs where the percentages using motorised private transport are highest and lowest.
   1. Using only the information in the diagram, suggest, with a reason, which of these two groups will have greater values of F than the other group. [1]
   A student says that it is not possible to give a reliable answer to part (b)(i) without some further information.
   1. Suggest two kinds of further information which would enable a more reliable answer to be given. [2]
3. Points \(A\) and \(B\) in the diagram are the most extreme outliers. Use their positions on the diagram to answer the following questions about the two LAs represented by these two points.
   1. The two LAs share a certain characteristic. Describe, with a justification, this characteristic. [2]
   2. The environments in these two LAs are very different. Describe, with a justification, this difference. [2]
4. A student says that it is difficult to extract detailed information from the scatter diagram. Explain whether you agree with this criticism. [1]