QUESTION (18 Marks) Two fishing trawlers return to Port Campbell after a weekend of fishing, jointly catching a total of 5000 fish. The number of fish caught by variety and trawler on this particular weekend is given below. Number of Tuna Number of Salmon Total Catch Boat 1 1800 900 3000 Boat 800 300 000 a. (i) A fish is selected at random from Boat 1. What is the probability that it is a tuna? (ii) 500 fish are selected at random from Boat 1. What is the expected number of tuna in this sample? b. (i) A fish is selected at random from all of the fish that were caught by both trawlers, and was found to be a tuna. What is the probability that this fish was caught by Boat 1? (ii) One fish is randomly selected from each boat. What is the probability that one of these fish is a tuna? Give your answer correct to three decimal places. marks The School For Excellence 017 Mathematical Methods Examination Questions Page 1
c. As each boat comes in, the catch is inspected and sorted according to type. All tuna are sorted into one bin and weighed. The salmon are sorted into their own bin, and the other varieties of fish are placed into a common bin. The fishermen are paid $10 per kilo for each kilogram of tuna caught. Salmon are paid at a rate of $1 per kilo, whereas the other varieties of fish are paid at a rate of $5 per kilogram. (i) Draw a probability distribution graph for the price paid per variety of fish to the fishermen on Boat 1. (ii) Hence calculate the mean price paid per kilogram to the fishermen on Boat 1. Give your answer to the nearest cent. 3 marks The School For Excellence 017 Mathematical Methods Examination Questions Page
d. (i) The weight of tuna delivered to Port Campbell is normally distributed with a mean of 35 kg. The fish are classified as acceptable for sale to John West Pty Ltd if the weight of a tuna lies within a units either side of the mean. The probability that a tuna from Port Campbell is accepted by John West is. Find an expression for a in terms of, correct to 4 decimal places. 0.70 3 marks (ii) A sample of 5 tuna is randomly selected from Boat 1. Find the probability that more than fish will be accepted by John West. Give your answer to 4 decimal places. marks The School For Excellence 017 Mathematical Methods Examination Questions Page 3
e. On another weekend, John West decided to inspect the tuna from every boat and trawler that docked at a rival port in Western Australia. The total number of tuna that had been caught that weekend was very large. John West randomly selected a total of 5 tuna from the entire weekend s catch at that port. p Let represent the proportion of tuna in the pool of fish caught over the weekend that fall within John West s acceptable weight range. (i) Write down an expression for the probability that of the 5 fish selected are within John West s acceptable weight range. (ii) Using calculus, find the value of p for which this probability will be a maximum. 3 marks Total Marks = 18 The School For Excellence 017 Mathematical Methods Examination Questions Page 4
Include the yellow highlighted sections in your exam response. QUESTION 1800 3 a. (i) Pr( tuna from Boat1) 3000 5 3 (ii) Let Number of tuna. E ( X ) 500 300 5 X b. (i) Pr( Boat1tuna) Pr( Boat1/ tuna) Pr( tuna) 1800 1800 9 5000 600 600 13 5000 Alternatively the probability may be obtained directly from the given table. Number tuna caught by Boat1 Total numbertuna caught 1800 600 9 13 (ii) Pr( oneis a tuna) Pr( tuna from Boat1 AND not from Boat ) OR Pr( tuna from Boat AND not from Boat1) 1800 100 3000 000 800 000 100 0.36 0.16 0.50 3000 c. (i) Tuna Salmon Other Number of fish 1800 900 300 Price per kilo (x$) 10 1 5 Probability (Pr(X=x) 1800 0.6 900 300 0. 3 0. 1 3000 3000 3000 Pr(X=x) 0.6 0.3 0.1 5 10 1 Price (x$) A3 The School For Excellence 017 Mathematical Methods Examination Questions Page 5
(ii) E($) (100.6) (10.3) (5 0.1) $10. 10 / kg d. (i) Area = 0.70 X Pr( X 35 a) 0.85 Pr( z c) 0.85 Using Inverse Normal: c 1.0364334 c 1. 0364 X 1.0364334 35 a 35 1.0364334 a 1.0364 (ii) Binomial. X Number acceptable fish p probabilit y acceptable 0.70 n 5 Find Pr( X ) 0.8369 0.8369 e. (i) Let X number of acceptable fish p p n 5 z c Pr( X ) 5 3 3 p 1 p 10p (1 p ) (ii) 3 P' ( p) 10 p 3(1 p) (1 p) 0 p 10p(1 p) ( 5p) Let P '( p) 0 : and simplify 10p (1 p) ( 5p) 0 p 0, 1, 5 As 0 p 1, p. 5 The School For Excellence 017 Mathematical Methods Examination Questions Page 6