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Problem 2657
2657
Last Saturday, Sarah recorded the colour and type of 160 cars in a car park. All the cars that were not red or silver in colour were grouped together as 'other'. Her results are shown in the following table.
Colour of car
Saloon
Hatchback
Estate
Red
20
40
12
Silver
14
26
10
Other
6
24
8
Find the probability that a randomly chosen car in the car park is a silver estate car. [1]
Find the probability that a randomly chosen car in the car park is a hatchback car. [1]
Find the probability that a randomly chosen car in the car park is red, given that it is a hatchback car. [2]
One of the cars in the car park is chosen at random. Determine whether the events 'the car is a hatchback car' and 'the car is red' are independent, justifying your answer. [2]
Solution
(i) The number of silver estate cars is 10. The total number of cars is 160. The probability is \(\frac{10}{160} = \frac{1}{16} = 0.0625\).
(ii) The total number of hatchback cars is \(40 + 26 + 24 = 90\). The probability is \(\frac{90}{160} = \frac{9}{16} = 0.5625\).
(iii) The number of red hatchback cars is 40. The probability of a car being a hatchback is \(\frac{90}{160}\). The probability of a car being red given it is a hatchback is \(\frac{40}{90} = \frac{4}{9}\).
(iv) For independence, \(P(\text{red}) \times P(\text{hatchback}) = \frac{72}{160} \times \frac{90}{160}\) should equal \(\frac{40}{160}\). Calculating, \(\frac{72}{160} \times \frac{90}{160} \neq \frac{40}{160}\), so the events are not independent.