CAIE S1 (Statistics 1) 2016 November

1 A committee of 5 people is to be chosen from 4 men and 6 women. William is one of the 4 men and Mary is one of the 6 women. Find the number of different committees that can be chosen if William and Mary refuse to be on the committee together.

2 A fair triangular spinner has three sides numbered 1, 2, 3. When the spinner is spun, the score is the number of the side on which it lands. The spinner is spun four times.

1. Find the probability that at least two of the scores are 3 .
2. Find the probability that the sum of the four scores is 5 .

3 Numbers are formed using some or all of the digits 4, 5, 6, 7 with no digit being used more than once.

1. Show that, using exactly 3 of the digits, there are 12 different odd numbers that can be formed.
2. Find how many odd numbers altogether can be formed.

4 For a group of 250 cars the numbers, classified by colour and country of manufacture, are shown in the table. One car is selected at random from this group. Find the probability that the selected car is

1. a red or silver car manufactured in Korea,
2. not manufactured in Japan.
   \(X\) is the event that the selected car is white. \(Y\) is the event that the selected car is manufactured in Germany.
3. By using appropriate probabilities, determine whether events \(X\) and \(Y\) are independent.

5 The tables summarise the heights, \(h \mathrm {~cm}\), of 60 girls and 60 boys.

1. On graph paper, using the same set of axes, draw two cumulative frequency graphs to illustrate the data.
2. On a school trip the students have to enter a cave which is 165 cm high. Use your graph to estimate the percentage of the girls who will be unable to stand upright.
   [0pt]
3. The students are asked to compare the heights of the girls and the boys. State one advantage of using a pair of box-and-whisker plots instead of the cumulative frequency graphs to do this. [1]

6 The weights of bananas in a fruit shop have a normal distribution with mean 150 grams and standard deviation 50 grams. Three sizes of banana are sold. Small: under 95 grams
Medium: between 95 grams and 205 grams
Large: over 205 grams

1. Find the proportion of bananas that are small.
2. Find the weight exceeded by \(10 \%\) of bananas. The prices of bananas are 10 cents for a small banana, 20 cents for a medium banana and 25 cents for a large banana.
3. (a) Show that the probability that a randomly chosen banana costs 20 cents is 0.7286 .
   (b) Calculate the expected total cost of 100 randomly chosen bananas.

7 Each day Annabel eats rice, potato or pasta. Independently of each other, the probability that she eats rice is 0.75 , the probability that she eats potato is 0.15 and the probability that she eats pasta is 0.1 .

1. Find the probability that, in any week of 7 days, Annabel eats pasta on exactly 2 days.
2. Find the probability that, in a period of 5 days, Annabel eats rice on 2 days, potato on 1 day and pasta on 2 days.
3. Find the probability that Annabel eats potato on more than 44 days in a year of 365 days.