CAIE S1 (Statistics 1) 2021 November

1 The 26 members of the local sports club include Mr and Mrs Khan and their son Abad. The club is holding a party to celebrate Abad's birthday, but there is only room for 20 people to attend. In how many ways can the 20 people be chosen from the 26 members of the club, given that Mr and Mrs Khan and Abad must be included?

2 Lakeview and Riverside are two schools. The pupils at both schools took part in a competition to see how far they could throw a ball. The distances thrown, to the nearest metre, by 11 pupils from each school are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Lakeview on the left-hand side.
2. Find the interquartile range of the distances thrown by the 11 pupils at Lakeview school.

3 The times taken, in minutes, by 360 employees at a large company to travel from home to work are summarised in the following table.

1. Draw a histogram to represent this information.
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2. Calculate an estimate of the mean time taken by an employee to travel to work.

4 Raj wants to improve his fitness, so every day he goes for a run. The times, in minutes, of his runs have a normal distribution with mean 41.2 and standard deviation 3.6.

1. Find the probability that on a randomly chosen day Raj runs for more than 43.2 minutes.
2. Find an estimate for the number of days in a year ( 365 days) on which Raj runs for less than 43.2 minutes.
3. On 95\% of days, Raj runs for more than \(t\) minutes. Find the value of \(t\).

5 A security code consists of 2 letters followed by a 4-digit number. The letters are chosen from \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) and the digits are chosen from \(\{ 1,2,3,4,5,6,7 \}\). No letter or digit may appear more than once. An example of a code is BE 3216 .

1. How many different codes can be formed?
2. Find the number of different codes that include the letter A or the digit 5 or both.
   A security code is formed at random.
3. Find the probability that the code is DE followed by a number between 4500 and 5000 .

6 In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye. The random variable \(X\) is the number of darts in a turn that hit the bull's-eye. The probability distribution of \(X\) is given in the following table. It is given that \(\mathrm { E } ( X ) = 0.55\).

1. Find the values of \(p\) and \(q\).
2. Find \(\operatorname { Var } ( X )\).
   Jim is practising for a competition and he repeatedly throws three darts at the board.
3. Find the probability that \(X = 1\) in at least 3 of 12 randomly chosen turns.
4. Find the probability that Jim first succeeds in hitting the bull's-eye with all three darts on his 9th turn.

7 Box \(A\) contains 6 red balls and 4 blue balls. Box \(B\) contains \(x\) red balls and 9 blue 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. Complete the tree diagram below, giving the remaining four probabilities in terms of \(x\).
   \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-12_688_759_484_731}
2. Show that the probability that both balls chosen are blue is \(\frac { 4 } { x + 10 }\).
   It is given that the probability that both balls chosen are blue is \(\frac { 1 } { 6 }\).
3. Find the probability, correct to 3 significant figures, that the ball chosen from box \(A\) is red given that the ball chosen from box \(B\) is red.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.