CAIE S1 (Statistics 1) 2013 November

1 It is given that \(X \sim \mathrm {~N} \left( 1.5,3.2 ^ { 2 } \right)\). Find the probability that a randomly chosen value of \(X\) is less than - 2.4 .

2 On Saturday afternoons Mohit goes shopping with probability 0.25, or goes to the cinema with probability 0.35 or stays at home. If he goes shopping the probability that he spends more than \(
) 50\( is 0.7 . If he goes to the cinema the probability that he spends more than \)\\( 50\) is 0.8 . If he stays at home he spends \(
) 10$ on a pizza.

1. Find the probability that Mohit will go to the cinema and spend less than \(
   ) 50\(.
2. Given that he spends less than \)\\( 50\), find the probability that he went to the cinema.

3 The amount of fibre in a packet of a certain brand of cereal is normally distributed with mean 160 grams. 19\% of packets of cereal contain more than 190 grams of fibre.

1. Find the standard deviation of the amount of fibre in a packet.
2. Kate buys 12 packets of cereal. Find the probability that at least 1 of the packets contains more than 190 grams of fibre.

4 The following histogram summarises the times, in minutes, taken by 190 people to complete a race.
\includegraphics[max width=\textwidth, alt={}, center]{df246a50-157b-49f7-bba0-f9b86960b8b9-2_1210_1125_1251_513}

1. Show that 75 people took between 200 and 250 minutes to complete the race.
2. Calculate estimates of the mean and standard deviation of the times of the 190 people.
3. Explain why your answers to part (ii) are estimates.

5 On trains in the morning rush hour, each person is either a student with probability 0.36 , or an office worker with probability 0.22 , or a shop assistant with probability 0.29 or none of these.

1. 8 people on a morning rush hour train are chosen at random. Find the probability that between 4 and 6 inclusive are office workers.
2. 300 people on a morning rush hour train are chosen at random. Find the probability that between 31 and 49 inclusive are neither students nor office workers nor shop assistants.

6 The 11 letters of the word REMEMBRANCE are arranged in a line.

1. Find the number of different arrangements if there are no restrictions.
2. Find the number of different arrangements which start and finish with the letter M .
3. Find the number of different arrangements which do not have all 4 vowels ( \(\mathrm { E } , \mathrm { E } , \mathrm { A } , \mathrm { E }\) ) next to each other. 4 letters from the letters of the word REMEMBRANCE are chosen.
4. Find the number of different selections which contain no Ms and no Rs and at least 2 Es.

7 Rory has 10 cards. Four of the cards have a 3 printed on them and six of the cards have a 4 printed on them. He takes three cards at random, without replacement, and adds up the numbers on the cards.

1. Show that P (the sum of the numbers on the three cards is \(11 ) = \frac { 1 } { 2 }\).
2. Draw up a probability distribution table for the sum of the numbers on the three cards. Event \(R\) is 'the sum of the numbers on the three cards is 11 '. Event \(S\) is 'the number on the first card taken is a \(3 ^ { \prime }\).
3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.
4. Determine whether events \(R\) and \(S\) are exclusive. Justify your answer.