CAIE S1 (Statistics 1) 2020 November

1 Two ordinary fair dice, one red and the other blue, are thrown.
Event \(A\) is 'the score on the red die is divisible by 3 '.
Event \(B\) is 'the sum of the two scores is at least 9 '.

1. Find \(\mathrm { P } ( A \cap B )\).
2. Hence determine whether or not the events \(A\) and \(B\) are independent.

2 The probability that a student at a large music college plays in the band is 0.6. For a student who plays in the band, the probability that she also sings in the choir is 0.3 . For a student who does not play in the band, the probability that she sings in the choir is \(x\). The probability that a randomly chosen student from the college does not sing in the choir is 0.58 .

1. Find the value of \(x\).
   Two students from the college are chosen at random.
2. Find the probability that both students play in the band and both sing in the choir.

3 Kayla is competing in a throwing event. A throw is counted as a success if the distance achieved is greater than 30 metres. The probability that Kayla will achieve a success on any throw is 0.25 .

1. Find the probability that Kayla takes more than 6 throws to achieve a success.
2. Find the probability that, for a random sample of 10 throws, Kayla achieves at least 3 successes.

4 The random variable \(X\) takes each of the values \(1,2,3,4\) with probability \(\frac { 1 } { 4 }\). Two independent values of \(X\) are chosen at random. If the two values of \(X\) are the same, the random variable \(Y\) takes that value. Otherwise, the value of \(Y\) is the larger value of \(X\) minus the smaller value of \(X\).

1. Draw up the probability distribution table for \(Y\).
2. Find the probability that \(Y = 2\) given that \(Y\) is even.

5 The time in hours that Davin plays on his games machine each day is normally distributed with mean 3.5 and standard deviation 0.9.

1. Find the probability that on a randomly chosen day Davin plays on his games machine for more than 4.2 hours.
2. On 90\% of days Davin plays on his games machine for more than \(t\) hours. Find the value of \(t\).
3. Calculate an estimate for the number of days in a year ( 365 days) on which Davin plays on his games machine for between 2.8 and 4.2 hours.

6 The times, \(t\) minutes, taken by 150 students to complete a particular challenge are summarised in the following cumulative frequency table.

1. Draw a cumulative frequency graph to illustrate the data.
   \includegraphics[max width=\textwidth, alt={}, center]{033ceb76-8fd4-4a89-ab05-5e20039d1c8d-08_1689_1195_744_516}
2. \(24 \%\) of the students take \(k\) minutes or longer to complete the challenge. Use your graph to estimate the value of \(k\).
3. Calculate estimates of the mean and the standard deviation of the time taken to complete the challenge.

7

1. Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that all 3 Es are together.
2. Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that the Ps are not next to each other.
3. Find the probability that a randomly chosen arrangement of the 10 letters of the word SHOPKEEPER has an E at the beginning and an E at the end.
   Four letters are selected from the 10 letters of the word SHOPKEEPER.
4. Find the number of different selections if the four letters include exactly one P .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.