7 A committee of 7 people is to be chosen at random from 18 volunteers.
- 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
- consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester,
- include exactly 5 people from Worcester,
- include at least 2 people from each of the three cities.
1 Jenny and John are each allowed two attempts to pass an examination.
- 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. - 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.
- Describe briefly how the heights of the second group of plants compare with the first.
- 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.
- Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 5 }\).
- The probability distribution of \(X\) is given in the table below.
| \(x\) | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 10 }\) | \(\frac { 1 } { 5 }\) | \(\frac { 3 } { 10 }\) | \(\frac { 2 } { 5 }\) |
\includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-07_675_1529_452_349}
The histogram shows the results for 300 days. By considering the total area of the histogram,
(a) find the number of days on which the daily income was between \(\pounds 4000\) and \(\pounds 6000\),
(b) calculate an estimate of the number of days on which the daily income was between \(\pounds 2700\) and \(\pounds 3200\).- The Research Department offers to provide any of the following statistical diagrams: histogram, frequency polygon, box-and-whisker plot, cumulative frequency graph, stem-and-leaf diagram and pie chart.
Which one of these statistical diagrams would most easily enable managers to
(a) read off the median and quartile values of the daily income,
(b) find the range of the top \(10 \%\) of values of the daily income?
5 Andrea practises shots at goal. For each shot the probability of her scoring a goal is \(\frac { 2 } { 5 }\). Each shot is independent of other shots. - Find the probability that she scores her first goal
(a) on her 5th shot,
(b) before her 5th shot. - (a) Find the probability that she scores exactly 1 goal in her first 5 shots.
(b) Hence find the probability that she scores her second goal on her 6th shot.
6 An examination paper consists of two parts. Section A contains questions A1, A2, A3 and A4. Section B contains questions \(\mathrm { B } 1 , \mathrm {~B} 2 , \mathrm {~B} 3 , \mathrm {~B} 4 , \mathrm {~B} 5 , \mathrm {~B} 6\) and B 7 .
Candidates must choose three questions from section A and four questions from section B. The order in which they choose the questions does not matter. - In how many ways can the seven questions be chosen?
- Assuming that all selections are equally likely, find the probability that a particular candidate chooses question A1 but does not choose question B1.
- Following a change of syllabus, the form of the examination remains the same except that candidates who choose question A1 are not allowed to choose question B1. In how many ways can the seven questions now be chosen?
7 Past experience has shown that when seeds of a certain type are planted, on average \(90 \%\) will germinate. A gardener plants 10 of these seeds in a tray and waits to see how many will germinate.
- Name an appropriate distribution with which to model the number of seeds that germinate, giving the value(s) of any parameters. State any assumption(s) needed for the model to be valid.
- Use your model to find the probability that fewer than 8 seeds germinate.
Later the gardener plants 20 trays of seeds, with 10 seeds in each tray.
- Calculate the probability that there are at least 19 trays in each of which at least 8 seeds germinate.
\section*{Jan 2006}