CAIE S1 (Statistics 1) 2020 November

1 A fair six-sided die, with faces marked \(1,2,3,4,5,6\), is thrown repeatedly until a 4 is obtained.

1. Find the probability that obtaining a 4 requires fewer than 6 throws.
   On another occasion, the die is thrown 10 times.
2. Find the probability that a 4 is obtained at least 3 times.

2 A bag contains 5 red balls and 3 blue balls. Sadie takes 3 balls at random from the bag, without replacement. The random variable \(X\) represents the number of red balls that she takes.

1. Show that the probability that Sadie takes exactly 1 red ball is \(\frac { 15 } { 56 }\).
2. Draw up the probability distribution table for \(X\).
3. Given that \(\mathrm { E } ( X ) = \frac { 15 } { 8 }\), find \(\operatorname { Var } ( X )\).

3 Pia runs 2 km every day and her times in minutes are normally distributed with mean 10.1 and standard deviation 1.3.

1. Find the probability that on a randomly chosen day Pia takes longer than 11.3 minutes to run 2 km .
2. On \(75 \%\) of days, Pia takes longer than \(t\) minutes to run 2 km . Find the value of \(t\).
3. On how many days in a period of 90 days would you expect Pia to take between 8.9 and 11.3 minutes to run 2 km ?

4 In a certain country, the weather each day is classified as fine or rainy. The probability that a fine day is followed by a fine day is 0.75 and the probability that a rainy day is followed by a fine day is 0.4 . The probability that it is fine on 1 April is 0.8 . The tree diagram below shows the possibilities for the weather on 1 April and 2 April.

1. Complete the tree diagram to show the probabilities. 1 April
   \includegraphics[max width=\textwidth, alt={}, center]{33c0bd01-f27b-424c-a78a-6f36178bc08c-08_601_405_706_408} 2 April Fine Rainy Fine Rainy
2. Find the probability that 2 April is fine.
   Let \(X\) be the event that 1 April is fine and \(Y\) be the event that 3 April is rainy.
3. Find the value of \(\mathrm { P } ( X \cap Y )\).
4. Find the probability that 1 April is fine given that 3 April is rainy.

5 The following table gives the weekly snowfall, in centimetres, for 11 weeks in 2018 at two ski resorts, Dados and Linva.

1. Represent the information in a back-to-back stem-and-leaf diagram.
2. Find the median and the interquartile range for the weekly snowfall in Dados.
3. The median, lower quartile and upper quartile of the weekly snowfall for Linva are 12, 9 and 32 cm respectively. Use this information and your answers to part (b) to compare the central tendency and the spread of the weekly snowfall in Dados and Linva.

6 Mr and Mrs Ahmed with their two children, and Mr and Mrs Baker with their three children, are visiting an activity centre together. They will divide into groups for some of the activities.

1. In how many ways can the 9 people be divided into a group of 6 and a group of 3?
   5 of the 9 people are selected at random for a particular activity.
2. Find the probability that this group of 5 people contains all 3 of the Baker children.
   All 9 people stand in a line.
3. Find the number of different arrangements in which Mr Ahmed is not standing next to Mr Baker.
4. Find the number of different arrangements in which there is exactly one person between Mr Ahmed and Mr Baker.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.