CAIE S1 (Statistics 1) 2024 November

1 Nicola throws an ordinary fair six-sided dice. The random variable \(X\) is the number of throws that she takes to obtain a 6.

1. Find \(\mathrm { P } ( X < 8 )\).
2. Find the probability that Nicola obtains a 6 for the second time on her 8th throw.
   \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-02_2717_35_109_2012}

2 The random variable \(X\) takes the values \(- 2 , - 1,0,2,3\). It is given that \(\mathrm { P } ( X = x ) = k \left( x ^ { 2 } + 2 \right)\), where \(k\) is a positive constant.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find the value of \(\operatorname { Var } ( X )\).

3 The time taken, in minutes, to walk to school was recorded for 200 pupils at a certain school. These times are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data.
   \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-04_1217_1509_705_278}
2. Use your graph to estimate the median and the interquartile range of the data.
   \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-05_2723_35_101_20}
3. Calculate an estimate for the mean value of the times taken by the 200 pupils to walk to school.

4 Rahul has two bags, \(X\) and \(Y\). Bag \(X\) contains 4 red marbles and 2 blue marbles. Bag \(Y\) contains 3 red marbles and 4 blue marbles. Rahul also has a coin which is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). Rahul throws the coin.

• If he obtains a head, he chooses at random a marble from bag \(X\). He notes the colour and replaces the marble in bag \(X\). He then chooses at random a second marble from bag \(X\).
• If he obtains a tail, he chooses at random a marble from bag \(Y\). He notes the colour and discards the marble. He then chooses at random a second marble from bag \(Y\).
  1. Find the probability that the two marbles that Rahul chooses are the same colour.
     \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-06_2717_33_109_2014}
     \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-07_2725_35_99_20}
  2. Find the probability that the two marbles that Rahul chooses are both from bag \(Y\) given that both marbles are blue.
     

5 The weights of the green apples sold by a shop are normally distributed with mean 90 grams and standard deviation 8 grams.

1. Find the probability that a randomly chosen green apple weighs between 83 grams and 95 grams.
   \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-09_2717_29_105_22}
2. The shop also sells red apples. \(60 \%\) of the red apples sold by the shop weigh more than 80 grams. 160 red apples are chosen at random from the shop. Use a suitable approximation to find the probability that fewer than 105 of the chosen red apples weigh more than 80 grams.

6 The heights of the female students at Breven college are normally distributed:

• \(90 \%\) of the female students have heights less than 182.7 cm .
• \(40 \%\) of the female students have heights less than 162.5 cm .
  1. Find the mean and the standard deviation of the heights of the female students at Breven college.
     \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-10_2715_41_110_2008}
     \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-11_2723_35_101_20}

Ten female students are chosen at random from those at Breven college.

Find the probability that fewer than 8 of these 10 students have heights more than 162.5 cm .

7

1. How many different arrangements are there of the 9 letters in the word INTELLECT in which the two Ts are together?
2. How many different arrangements are there of the 9 letters in the word INTELLECT in which there is a T at each end and the two Es are not next to each other?
   Four letters are selected at random from the 9 letters in the word INTELLECT.
   [0pt]
3. Find the percentage of the possible selections which contain at least one E and exactly one T. [4]
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-14_2715_31_106_2016}