CAIE S1 (Statistics 1) 2009 November

1 The mean number of defective batteries in packs of 20 is 1.6 . Use a binomial distribution to calculate the probability that a randomly chosen pack of 20 will have more than 2 defective batteries.

2 The probability distribution of the random variable \(X\) is shown in the following table. The mean of \(X\) is 1.05 .

1. Write down two equations involving \(p\) and \(q\) and hence find the values of \(p\) and \(q\).
2. Find the variance of \(X\).

3 The times for a certain car journey have a normal distribution with mean 100 minutes and standard deviation 7 minutes. Journey times are classified as follows: \begin{displayquote} 'short' (the shortest \(33 \%\) of times),
'long' (the longest \(33 \%\) of times),
'standard' (the remaining 34\% of times).

1. Find the probability that a randomly chosen car journey takes between 85 and 100 minutes.
2. Find the least and greatest times for 'standard' journeys. \end{displayquote}

4 A library has many identical shelves. All the shelves are full and the numbers of books on each shelf in a certain section are summarised by the following stem-and-leaf diagram. Key: 3 | 6 represents 36 books

1. Find the number of shelves in this section of the library.
2. Draw a box-and-whisker plot to represent the data. In another section all the shelves are full and the numbers of books on each shelf are summarised by the following stem-and-leaf diagram.

   2	12222334566679	\(( 13 )\)
   3	01112334456677788	\(( 15 )\)
   4	223357789

   Key: 3 | 6 represents 36 books
3. There are fewer books in this section than in the previous section. State one other difference between the books in this section and the books in the previous section.

5

1. Find how many numbers between 5000 and 6000 can be formed from the digits 1, 2, 3, 4, 5 and 6
   1. if no digits are repeated,
   2. if repeated digits are allowed.
2. Find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls
   1. if there are more girls than boys in the team,
   2. if three of the boys are cousins and are either all in the team or all not in the team.

6 A box contains 4 pears and 7 oranges. Three fruits are taken out at random and eaten. Find the probability that

1. 2 pears and 1 orange are eaten, in any order,
2. the third fruit eaten is an orange,
3. the first fruit eaten was a pear, given that the third fruit eaten is an orange. There are 121 similar boxes in a warehouse. One fruit is taken at random from each box.
4. Using a suitable approximation, find the probability that fewer than 39 are pears.