CAIE S1 (Statistics 1) 2020 March

1 The 40 members of a club include Ranuf and Saed. All 40 members will travel to a concert. 35 members will travel in a coach and the other 5 will travel in a car. Ranuf will be in the coach and Saed will be in the car. In how many ways can the members who will travel in the coach be chosen?

2 An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.

1. Find the probability that it takes at least 3 throws but no more than 5 throws to obtain a 1 or a 6 .
   On another occasion, this die is thrown 3 times. The random variable \(X\) is the number of times that a 1 or a 6 is obtained.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

3 The weights of apples of a certain variety are normally distributed with mean 82 grams. \(22 \%\) of these apples have a weight greater than 87 grams.

1. Find the standard deviation of the weights of these apples.
2. Find the probability that the weight of a randomly chosen apple of this variety differs from the mean weight by less than 4 grams.

4 Richard has 3 blue candles, 2 red candles and 6 green candles. The candles are identical apart from their colours. He arranges the 11 candles in a line.

1. Find the number of different arrangements of the 11 candles if there is a red candle at each end.
2. Find the number of different arrangements of the 11 candles if all the blue candles are together and the red candles are not together.

5 In Greenton, 70\% of the adults own a car. A random sample of 8 adults from Greenton is chosen.
[0pt]

1. Find the probability that the number of adults in this sample who own a car is less than 6 . [3]
   A random sample of 120 adults from Greenton is now chosen.
2. Use an approximation to find the probability that more than 75 of them own a car.

6 Box \(A\) contains 7 red balls and 1 blue ball. Box \(B\) contains 9 red balls and 5 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 \(B\). The tree diagram below shows the possibilities for the colours of the balls chosen.

1. Complete the tree diagram to show the probabilities. Box \(A\)
   \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-08_624_428_667_621} \section*{Box \(B\)} Red Blue Red Blue
2. Find the probability that the two balls chosen are not the same colour.
3. Find the probability that the ball chosen from box \(A\) is blue given that the ball chosen from box \(B\) is blue.

7 Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data.
   \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-10_1593_1296_790_466}
2. 40\% of these fish have a length of \(d \mathrm {~cm}\) or more. Use your graph to estimate the value of \(d\).
   The mean length of these 150 fish is 15.295 cm .
3. Calculate an estimate for the variance of the lengths of the fish.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.