CAIE S1 (Statistics 1) 2023 November

1 Becky sometimes works in an office and sometimes works at home. The random variable \(X\) denotes the number of days that she works at home in any given week. It is given that $$\mathrm { P } ( X = x ) = k x ( x + 1 )$$ where \(k\) is a constant and \(x = 1,2,3\) or 4 only.

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

2 The weights of large bags of pasta produced by a company are normally distributed with mean 1.5 kg and standard deviation 0.05 kg .

1. Find the probability that a randomly chosen large bag of pasta weighs between 1.42 kg and 1.52 kg .
   The weights of small bags of pasta produced by the company are normally distributed with mean 0.75 kg and standard deviation \(\sigma \mathrm { kg }\). It is found that \(68 \%\) of these small bags have weight less than 0.9 kg .
2. Find the value of \(\sigma\).

3 Tim has two bags of marbles, \(A\) and \(B\).
Bag \(A\) contains 8 white, 4 red and 3 yellow marbles.
Bag \(B\) contains 6 white, 7 red and 2 yellow marbles.
Tim also has an ordinary fair 6 -sided dice. He rolls the dice. If he obtains a 1 or 2 , he chooses two marbles at random from bag \(A\), without replacement. If he obtains a \(3,4,5\) or 6 , he chooses two marbles at random from bag \(B\), without replacement.

1. Find the probability that both marbles are white.
2. Find the probability that the two marbles come from bag \(B\) given that one is white and one is red. [4]

4 The weights, \(x \mathrm {~kg}\), of 120 students in a sports college are recorded. The results are summarised in the following table.

1. Draw a cumulative frequency graph to represent this information.
   \includegraphics[max width=\textwidth, alt={}, center]{82c36c11-878c-47d1-a07f-fbf8b2a22d97-06_1390_1389_660_418}
2. It is found that \(35 \%\) of the students weigh more than \(W \mathrm {~kg}\). Use your graph to estimate the value of \(W\).
3. Calculate estimates for the mean and standard deviation of the weights of the 120 students. [6]

5 The probability that a driver passes an advanced driving test is 0.3 on any given attempt.

1. Dipak keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for Dipak to pass the test.
   1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 6 )\).
   2. Find \(\mathrm { E } ( X )\).
      Five friends will each take their advanced driving test tomorrow.
2. Find the probability that at least three of them will pass tomorrow.
   75 people will take their advanced driving test next week.
   [0pt]
3. Use an approximation to find the probability that more than 20 of them will pass next week. [5]

6 Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife.

1. At the beginning of the party, the 12 people will stand in a line for a photograph.
   1. How many different arrangements are there of the 12 people if Jai stands next to Kaz and each friend stands next to his own wife?
   2. How many different arrangements are there of the 12 people if Jai and Kaz occupy the two middle positions in the line, with Jai's five friends on one side and the five wives of the friends on the other side?
2. For a competition during the party, the 12 people are divided at random into a group of 5, a group of 4 and a group of 3 . Find the probability that Jai and Kaz are in the same group as each other.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.