CAIE S1 (Statistics 1) 2024 November

2 A red fair six-sided dice has faces labelled 1, 1, 1, 2, 2, 2. A blue fair six-sided dice has faces labelled \(1,1,2,2,3,3\). Both dice are thrown. The random variable \(X\) is the product of the scores on the two dice.

1. Draw up the probability distribution table for \(X\).
2. Find \(\mathrm { E } ( X )\).

3 In Molimba, the heights, in cm , of adult males are normally distributed with mean 176 cm and standard deviation 4.8 cm .

1. Find the probability that a randomly chosen adult male in Molimba has a height greater than 170 cm .
   60\% of adult males in Molimba have a height between 170 cm and \(k \mathrm {~cm}\), where \(k\) is greater than 170 .
2. Find the value of \(k\), giving your answer correct to 1 decimal place.

4 On a certain day, the heights of 150 sunflower plants grown by children at a local school are measured, correct to the nearest cm . These heights are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data.
   \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-06_1600_1301_760_383}
2. Use your graph to estimate the 30th percentile of the heights of the sunflower plants.
   \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-07_2723_35_101_20}
3. Calculate estimates for the mean and the standard deviation of the heights of the 150 sunflower plants.

5 A factory produces chocolates. 30\% of the chocolates are wrapped in gold foil, 25\% are wrapped in red foil and the remainder are unwrapped. Indigo chooses 8 chocolates at random from the production line.

1. Find the probability that she obtains no more than 2 chocolates that are wrapped in red foil.
   Jake chooses chocolates one at a time at random from the production line.
2. Find the probability that the first time he obtains a chocolate that is wrapped in red foil is before the 7th choice.
   \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-08_2720_35_106_2015}
   \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-09_2717_29_105_22} Keifa chooses chocolates one at a time at random from the production line.
3. Find the probability that the second chocolate chosen is the first one wrapped in gold foil given that the fifth chocolate chosen is the first unwrapped chocolate.

6

1. Find the number of different arrangements of the 9 letters in the word HAPPINESS.
2. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the first and last letters are not the same as each other.
   \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-10_2715_35_110_2012}
3. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the two Ps are together and there are exactly two letters between the two Ss.
   The 9 letters in the word HAPPINESS are divided at random into a group of 5 and a group of 4 .
4. Find the probability that both Ps are in one group and both Ss are in the other group.
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