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MATHEMATICAL METHODS UNIT 4 School Assessed Coursework 3 (SAC 3): Problem-Solving Task 2 (Probability and Statistics)

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MATHEMATICAL METHODS UNIT 4 School Assessed Coursework 3 (SAC 3): Problem-Solving Task 2 (Probability and Statistics) Question 1 (47 marks) Galaxian Highscore, the intrepid adventurer, buys all his strawberries in packets from the Moles supermarket. At Moles the number X of strawberries in a large packet is a discrete random variable with probability distribution as shown in the following table: x | 24 | 25 | 26 | 27 Pr( X=x) | 1 8 | 3 k2 | 3 8 | 14 +ka. Find all possible values of k. 3 marks Question 1 - continued PROBLEM SOLVING TASK Page 3 At the Moles supermarket the number Y of strawberries in a small packet is a discrete random variable with probability distribution as shown in the following table: y | 14 | 15 | 16 | 17 Pr(Y= y) | 0.21 | 0.38 | 0.23 | 0.18b. Galaxian buys two small packets of strawberries. Find the probability, correct to four decimal places, that i. each packet has 14 strawberries. 1 mark ii. the total number of strawberries in both packets is equal to 32. 2 marks iii. if the total number of strawberries in both packets is equal to 32 then one of the packets has 15 strawberries 3 marks Question 1 - continued TURN OVER PROBLEM SOLVING TASK Page 4 c. On another occasion Galaxian buys seven small packets of strawberries. Find the probability, correct to four decimal places, that i. at least three of the packets have 16 strawberries. 2 marks ii. if at least two of the packets have 16 strawberries then exactly four of the packets have 16 strawberries. 3 marks Question 1 - continued PROBLEM SOLVING TASK Page 5 Max Hardcase, the steadfast partner-in-adventure of Galaxian Highscore, also buys all his strawberries from the Moles supermarket. He buys a number of small packets of strawberries in readiness for a new adventure. d. i. Find the smallest number of packets that Max could have bought if the probability that at least 10 of his packets have less than 16 strawberries is greater than 0.9 3 marks A regular strawberry has a width of at least 2 cm. One of the small packets that Max buys has 15 strawberries in it and 12 of those strawberries are regular. ii. Max selects a random sample of 11 strawberries from this packet. Find the probability that more than 8 of the strawberries in the sample are regular. 3 marks Question 1 - continued TURN OVER PROBLEM SOLVING TASK Page 6 Moles supermarket claims that 90% of all the strawberries it sells are regular. Galaxian Highscore collects a random sample of 51 strawberries after buying two large packets. e. i. Find, correct to four decimal places, the probability that at least 44 of the strawberries in his sample are regular.

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