CAIE S1 (Statistics 1) 2024 June

1 The numbers on the faces of a fair six-sided dice are \(1,2,2,3,3,3\). The random variable \(X\) is the total score when the dice is rolled twice.

1. Draw up the probability distribution table for \(X\).
2. Find the value of \(\operatorname { Var } ( X )\).
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-02_2714_34_143_2012}
3. Find the probability that \(X\) is even given that \(X > 3\).

2 In a certain country, the heights of the adult population are normally distributed with mean 1.64 m and standard deviation 0.25 m .

1. Find the probability that an adult chosen at random from this country will have height greater than 1.93 m .
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-04_2716_35_143_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-05_2724_35_136_20} In another country, the heights of the adult population are also normally distributed. \(33 \%\) of the adult population have height less than \(1.56 \mathrm {~m} .25 \%\) of the adult population have height greater than 1.86 m .
2. Find the mean and the standard deviation of this distribution.

3 Box \(A\) contains 6 green balls and 3 yellow balls.
Box \(B\) contains 4 green balls and \(x\) yellow balls.
A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\).

1. Draw a tree diagram to represent this information, showing the probability on each of the branches.
   [0pt] [4]
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-06_2727_38_132_2010}
   The probability that both the balls chosen are the same colour is \(\frac { 8 } { 15 }\).
2. Find the value of \(x\).

4 The times taken, in seconds, by 15 members of each of two swimming clubs, the Penguins and the Dolphins, to swim 50 metres are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Penguins on the left-hand side.
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-09_2720_33_141_20} The diagram shows a box-and-whisker plot representing the times for the Penguins.
2. On the same diagram, draw a box-and-whisker plot to represent the times for the Dolphins.
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-09_719_1219_424_424}
3. Hence state one difference between the distributions of the times for the Penguins and the Dolphins.

5 Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.
[0pt]

1. Find the probability that Salah completes the puzzle for the first time on the 5th day. [1]
2. Find the probability that Salah completes the puzzle for the second time on the 5th day.
3. Find the probability that Salah completes the puzzle fewer than 5 times in a week (7 days). [3]
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-10_2713_31_145_2014}
4. Use a suitable approximation to find the probability that Salah completes the puzzle more than 50 times in a period of 84 days.

6

1. How many different arrangements are there of the 9 letters in the word RECORDERS?
2. How many different arrangements are there of the 9 letters in the word RECORDERS in which there is an E at the beginning, an E at the end and the three Rs are not all together?
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-12_2725_40_136_2007}
   The 9 letters of the word RECORDERS are divided at random into two groups: a group of 5 letters and a group of 4 letters.
3. Find the probability that the three Rs are in the same group.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-14_2715_35_143_2012}