CAIE S1 (Statistics 1) 2022 November

2 In a large college, \(32 \%\) of the students have blue eyes. A random sample of 80 students is chosen. Use an approximation to find the probability that fewer than 20 of these students have blue eyes.

3 The times, \(t\) minutes, taken to complete a walking challenge by 250 members of a club are summarised in the table.

1. Draw a cumulative frequency graph to illustrate the data.
   \includegraphics[max width=\textwidth, alt={}, center]{1eb957f4-5088-4991-aa8a-f895d55d2bcf-04_1395_1298_705_466}
2. Use your graph to estimate the 60th percentile of the data.
   It is given that an estimate for the mean time taken to complete the challenge by these 250 members is 34.4 minutes.
3. Calculate an estimate for the standard deviation of the times taken to complete the challenge by these 250 members.

4 Three fair 4-sided spinners each have sides labelled 1,2,3,4. The spinners are spun at the same time and the number on the side on which each spinner lands is recorded. The random variable \(X\) denotes the highest number recorded.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 64 }\).
2. Complete the probability distribution table for \(X\).

   \(x\)	1	2	3	4
   \(\mathrm { P } ( X = x )\)		\(\frac { 7 } { 64 }\)	\(\frac { 19 } { 64 }\)

   On another occasion, one of the fair 4 -sided spinners is spun repeatedly until a 3 is obtained. The random variable \(Y\) is the number of spins required to obtain a 3 .
3. Find \(\mathrm { P } ( Y = 6 )\).
4. Find \(\mathrm { P } ( Y > 4 )\).

5 Company \(A\) produces bags of sugar. An inspector finds that on average \(10 \%\) of the bags are underweight. 10 of the bags are chosen at random.

1. Find the probability that fewer than 3 of these bags are underweight.
   The weights of the bags of sugar produced by company \(B\) are normally distributed with mean 1.04 kg and standard deviation 0.06 kg .
2. Find the probability that a randomly chosen bag produced by company \(B\) weighs more than 1.11 kg .
   \(81 \%\) of the bags of sugar produced by company \(B\) weigh less than \(w \mathrm {~kg}\).
3. Find the value of \(w\).

6

1. Find the number of different arrangements of the 9 letters in the word ACTIVATED.
2. Find the number of different arrangements of the 9 letters in the word ACTIVATED in which there are at least 5 letters between the two As.
   Five letters are selected at random from the 9 letters in the word ACTIVATED.
3. Find the probability that the selection does not contain more Ts than As.

7 Sam and Tom are playing a game which involves a bag containing 5 white discs and 3 red discs. They take turns to remove one disc from the bag at random. Discs that are removed are not replaced into the bag. The game ends as soon as one player has removed two red discs from the bag. That player wins the game. Sam removes the first disc.

1. Find the probability that Tom removes a red disc on his first turn.
2. Find the probability that Tom wins the game on his second turn.
3. Find the probability that Sam removes a red disc on his first turn given that Tom wins the game on his second turn.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.