CAIE S1 (Statistics 1) 2011 June

1 A biased die was thrown 20 times and the number of 5 s was noted. This experiment was repeated many times and the average number of 5 s was found to be 4.8 . Find the probability that in the next 20 throws the number of 5 s will be less than three.

2 In Scotland, in November, on average \(80 \%\) of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.

1. Use a normal approximation to find the probability of there being fewer than 25 cloudy days in Scotland in November (30 days).
2. Give a reason why the use of a normal approximation is justified.

3 A sample of 36 data values, \(x\), gave \(\Sigma ( x - 45 ) = - 148\) and \(\Sigma ( x - 45 ) ^ { 2 } = 3089\).

1. Find the mean and standard deviation of the 36 values.
2. One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.

4

1. Find the number of different ways that the 9 letters of the word HAPPINESS can be arranged in a line.
2. The 9 letters of the word HAPPINESS are arranged in random order in a line. Find the probability that the 3 vowels (A, E, I) are not all next to each other.
3. Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS which contain no Ps and either one or two Ss.

5 A hotel has 90 rooms. The table summarises information about the number of rooms occupied each day for a period of 200 days.

1. Draw a cumulative frequency graph on graph paper to illustrate this information.
2. Estimate the number of days when over 30 rooms were occupied.
3. On \(75 \%\) of the days at most \(n\) rooms were occupied. Estimate the value of \(n\).

6 The lengths, in centimetres, of drinking straws produced in a factory have a normal distribution with mean \(\mu\) and variance 0.64 . It is given that \(10 \%\) of the straws are shorter than 20 cm .

1. Find the value of \(\mu\).
2. Find the probability that, of 4 straws chosen at random, fewer than 2 will have a length between 21.5 cm and 22.5 cm .

7 Judy and Steve play a game using five cards numbered 3, 4, 5, 8, 9. Judy chooses a card at random, looks at the number on it and replaces the card. Then Steve chooses a card at random, looks at the number on it and replaces the card. If their two numbers are equal the score is 0 . Otherwise, the smaller number is subtracted from the larger number to give the score.

1. Show that the probability that the score is 6 is 0.08 .
2. Draw up a probability distribution table for the score.
3. Calculate the mean score. If the score is 0 they play again. If the score is 4 or more Judy wins. Otherwise Steve wins. They continue playing until one of the players wins.
4. Find the probability that Judy wins with the second choice of cards.
5. Find an expression for the probability that Judy wins with the \(n\)th choice of cards.