CAIE S1 (Statistics 1) 2024 November

1 At a college, the students choose exactly one of tennis, hockey or netball to play. The table shows the numbers of students in Year 1 and Year 2 at the college playing each of these sports. One student is chosen at random from the 120 students. Events \(X\) and \(N\) are defined as follows:
\(X\) : the student is in Year 1
\(N\) : the student plays netball.

1. Find \(\mathrm { P } ( X \mid N )\).
2. Find \(\mathrm { P } ( N \mid X )\).
3. Determine whether or not \(X\) and \(N\) are independent events.
   One of the students who plays netball takes 8 shots at goal. On each shot, the probability that she will succeed is 0.15 , independently of all other shots.
4. Find the probability that she succeeds on fewer than 3 of these shots.

2

1. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC.
2. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC in which there are no more than two letters between the two As.
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3 A fair coin and an ordinary fair six-sided dice are thrown at the same time.The random variable \(X\) is defined as follows.
-If the coin shows a tail,\(X\) is twice the score on the dice.
-If the coin shows a head,\(X\) is the score on the dice if the score is even and \(X\) is 0 otherwise.

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

4 The heights, in metres, of white pine trees are normally distributed with mean 19.8 and standard deviation 2.4 . In a certain forest there are 450 white pine trees.

1. How many of these trees would you expect to have height less than 18.2 metres?
   The heights, in metres, of red pine trees are normally distributed with mean 23.4 and standard deviation \(\sigma\). It is known that \(26 \%\) of red pine trees have height greater than 25.5 metres.
2. Find the value of \(\sigma\).

5 In a class of 21 students, there are 10 violinists, 6 guitarists and 5 pianists. A group of 7 is to be chosen from these 21 students. The group will consist of 4 violinists, 2 guitarists and 1 pianist.

1. In how many ways can the group of 7 be chosen?
   On another occasion a group of 5 will be chosen from the 21 students. The group must contain at least 2 violinists, at least 1 guitarist and at most 1 pianist.
2. In how many ways can the group of 5 be chosen?

6 Teams of 15 runners took part in a charity run last Saturday. The times taken, in minutes, to complete the course by the runners from the Falcons and the runners from the Kites are shown in the table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Falcons on the left-hand side.
2. Find the median and the interquartile range of the times for the Falcons.
   Let \(x\) and \(y\) denote the times, in minutes, of a runner from the Falcons and a runner from the Kites respectively. It is given that $$\sum x = 792 , \quad \sum x ^ { 2 } = 43504 , \quad \sum y = 783 , \quad \sum y ^ { 2 } = 42223 .$$
3. Find the mean and the standard deviation of the times taken by all 30 runners from the two teams.

7 In a game,players attempt to score a goal by kicking a ball into a net.The probability that Leno scores a goal is 0.4 on any attempt,independently of all other attempts.The random variable \(X\) denotes the number of attempts that it takes Leno to score a goal.

1. Find \(\mathrm { P } ( X = 5 )\) .
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2. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\) .
3. Find the probability that Leno scores his second goal on or before his 5th attempt.
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   \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-11_2723_33_99_22} Leno has 75 attempts to score a goal.
4. Use a suitable approximation to find the probability that Leno scores more than 28 goals but fewer than 35 goals.
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