Statistics & Probability W1

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Mathematics HL STATISTICS & PROBABILITY Worksheet 1 1. No GDC Consider the data set { k − 2, k , k +1, k + 4}, where k . (a) Find the mean of this data set in terms of k . (3) Each number in the above data set is now decreased by 3. (b) Find the mean of this new data set in terms of k . (2) (Total 5 marks) 2. Events A and B are such that P( A ) = 0.3 and P( B ) = 0.4. (a) Find the value of P( A B ) when (i) A and B are mutually exclusive; (ii) A and B are independent. (4) (b) Given that P( A B ) = 0.6, find P( A | B ) . (3) (Total 7 marks) 3. In a population of rabbits, 1 % are known to have a particular disease. A test is developed for the disease that gives a positive result for a rabbit that does have the disease in 99 % of cases. It is also known that the test gives a positive result for a rabbit that does not have the disease in 0.1 % of cases. A rabbit is chosen at random from the population. (a) Find the probability that the rabbit tests positive for the disease. (2) (b) Given that the rabbit tests positive for the disease, show that the probability that the rabbit does not have the disease is less than 10 %. (3) (Total 5 marks) 4. The random variable X has probability density function f where f ( x ) = (a) Sketch the graph of the function. You are not required to find the coordinates of the maximum. (1) (b) Find the value of k. (5) (Total 6 marks) 5. A continuous random variable X has a probability density function given by the function f ( x ), where f ( x ) = (a) Find the value of k. (2) + otherwise. , 0 2 0 ), 2 )( 1 ( x x x kx + otherwise. , 0 3 4 0 , 0 2 ) 2 ( 2 x k x x k
(b) Hence find (i) the mean of X ; (ii) the median of X. (5) (Total 7 marks) 6. A student arrives at a school X minutes after 08:00, where X may be assumed to be normally distributed. On a particular day it is observed that 40 % of the students arrive before 08:30 and 90 % arrive before 08:55. (a) Find the mean and standard deviation of X. (5) (b) The school has 1200 students and classes start at 09:00. Estimate the number of students who will be late on that day. (3) (c) Maelis had not arrived by 08:30. Find the probability that she arrived late. (2) (Total 10 marks) 7. The fish in a lake have weights that are normally distributed with a mean of 1.3 kg and a standard deviation of 0.2 kg. (a) Determine the probability that a fish that is caught weighs less than 1.4 kg. (1) (b) John catches 6 fish. Calculate the probability that at least 4 of the fish weigh more than 1.4 kg. (3) (c) Determine the probability that a fish that is caught weighs less than 1 kg, given that it weighs less than 1.4 kg. (2) (Total 6 marks) 8. No GDC Two players, A and B, alternately throw a fair six - sided dice, with A starting, until one of them obtains a six. Find the probability that A obtains the first six. (Total 7 marks)
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