CAIE S1 (Statistics 1) 2020 November

1 The times taken to swim 100 metres by members of a large swimming club have a normal distribution with mean 62 seconds and standard deviation 5 seconds.

1. Find the probability that a randomly chosen member of the club takes between 56 and 66 seconds to swim 100 metres.
2. \(13 \%\) of the members of the club take more than \(t\) minutes to swim 100 metres. Find the value of \(t\).

2 An ordinary fair die is thrown until a 6 is obtained.

1. Find the probability that obtaining a 6 takes more than 8 throws.
   Two ordinary fair dice are thrown together until a pair of 6s is obtained. The number of throws taken is denoted by the random variable \(X\).
2. Find the expected value of \(X\).
3. Find the probability that obtaining a pair of 6s takes either 10 or 11 throws.

3 A committee of 6 people is to be chosen from 9 women and 5 men.

1. Find the number of ways in which the 6 people can be chosen if there must be more women than men on the committee.
   The 9 women and 5 men include a sister and brother.
2. Find the number of ways in which the committee can be chosen if the sister and brother cannot both be on the committee.

4 The 1300 train from Jahor to Keman runs every day. The probability that the train arrives late in Keman is 0.35 .

1. For a random sample of 7 days, find the probability that the train arrives late on fewer than 3 days.
   A random sample of 142 days is taken.
2. Use an approximation to find the probability that the train arrives late on more than 40 days.

5 The 8 letters in the word RESERVED are arranged in a random order.

1. Find the probability that the arrangement has V as the first letter and E as the last letter.
2. Find the probability that the arrangement has both Rs together given that all three Es are together.

6 Three coins \(A , B\) and \(C\) are each thrown once.

• Coins \(A\) and \(B\) are each biased so that the probability of obtaining a head is \(\frac { 2 } { 3 }\).
• Coin \(C\) is biased so that the probability of obtaining a head is \(\frac { 4 } { 5 }\).
  1. Show that the probability of obtaining exactly 2 heads and 1 tail is \(\frac { 4 } { 9 }\).
     

The random variable \(X\) is the number of heads obtained when the three coins are thrown.

Draw up the probability distribution table for \(X\).

Given that \(\mathrm { E } ( X ) = \frac { 32 } { 15 }\), find \(\operatorname { Var } ( X )\).

7 A particular piece of music was played by 91 pianists and for each pianist, the number of incorrect notes was recorded. The results are summarised in the table.

1. Draw a histogram to represent this information.
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2. State which class interval contains the lower quartile and which class interval contains the upper quartile. Hence find the greatest possible value of the interquartile range.
3. Calculate an estimate for the mean number of incorrect notes.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.