Mathematical Methods: Practice Problem Solving Task - Probability

Transcription

1 Mathematical Methods: Practice Problem Solving Task - Probability Question 1 refers to the following graph The following graph shows the probabilities of the 5 outcomes (1 to 5) from a spinner, with one missing. Question 1 (a) The missing entry on the graph would have a probability of:! A. " B. 3 C. 0.3 # D. $" E. 0.2 Show that the answer is D. 1

2 What incorrect reasoning would lead a student to choosing A as the answer? (1 mark) (b) Hence show the expected value of the number spun in the long run is (c) Ms Shardlow has tried to work out the variance for the number showing on this spinner. Unfortunately, she has made a mistake in her calculations. Her working is given below. Identify the error, circle it and then find the correct value for the variance. E(number on spinner) = 3.38 x $ x $ Pr (X = x) Variance = = (d) Find the Pr (μ σ x μ + σ) 2

3 (e) The median value and the modal value for this spinner will be, respectively: A. Median = 3.5, Mode = 4 B. Median = 3, Mode = 0.3 C. Median = 0.5, Mode = 0.3 D. Median = 4, Mode = 4 E. Median = 3, Mode = 4 Correct Response: Working and/or reasoning for your response: (3 marks) Question 2 Mr Schmidt plays basketball for his local team. A record of his performance over the last season was kept and part of this is represented in the Venn diagram below. n ε = 200 A = {goals scored} B = {attempts at a 3-point shot} (a) What is the probability that, given he scored a goal, it was a 3-pointer? A. 9$ :" B. $ # C. 0.3 D. $! 9; E Correct Response: 3

4 Working and/or reasoning for your response: (3 marks) (b) Determine whether events A and B are mathematically independent (c) (i) Let X = number of goals Mr Schmidt scores in 8 shots at goal We can write X Bi (8,!9# $;; ). The following mathematical expression gives the probability for what possible event? Describe the event in words. Pr (event) = 1 8 0!9# $;; ; "> $;;? + 8 1!9# $;;! "> $;; # ( 1 mark) (ii) Find the probability that Mr Schmidt was able to score the first 4 goals only in his 8 shots. Give your answer correct to 4 decimal places. 4

5 Question 3: The diagram below shows the graphs of two normal distribution curves with means of µ 1 and µ 2 and standard deviations of σ 1 and σ 2 respectively. X! ~N(μ!, σ! $ ) X $ ~N(μ $, σ $ $ ) (a) Which one of the following is true? A. µ 1 < µ 2 and σ 1 < σ 2 B. mode 1 < mode 2 and σ 1 > σ 2 C. µ 1 > µ 2 and σ 1 < σ 2 D. median 1 = median 2 and σ 1 > σ 2 E. µ 1 = µ 2 and σ 1 < σ 2 Correct Response: Working and/or reasoning: 5

6 (b) Sketch two normal distribution curves that would make Option B in (a) a correct response. Be careful to label your curves with X 1 and X 2 Question 4 (a) If X is a discrete variable with E(X) = 3 and VAR (X) = 25 then E ( 2X + 1) and VAR ( 2X + 1) are respectively: A. 3 and 25 B. 5 and 50 C. 7 and 51 D. 5 and 100 E. 5 and 100 Correct Response: Working/Reasoning: (3 marks) (b) Find the standard deviation of (3! $ X) 6

7 Question 5 Mrs Batsakis has two favouriate café she likes to visit for coffee, Life on Mars and Tony s Cafe. Over the holidays, she decided to visit only one of these cafes on any one day. If she visited Life on Mars one day, then the probability that she visited Tony s Cafe the next was 0.7. If she visited Tony s Cafe one day, then the probability that she will again visit Tony s Cafe on the next day is (a) This information can be summarised in the incomplete transition matrix, T, as Explain why the missing elements are 0.3 and 0.75 respectively and hence write down the probability that if Mrs Batsakis visited Tony s Cafe on one day, she will visit Life on Mars the next. (b) Find the probability, giving your answer correct to 3 decimal places, that if Mrs Batsakis visited Tony s Café on a Monday, that she would be visiting Tony s Café again on the Friday of that same week. 7

8 (c) The steady state probability of Mrs Batsakis visiting Life on Mars Cafe on a given day if she was to continue this pattern of behaviour through until the end of the year would be A.!" $: B.!9 $: C. > 9 D.! 9 E. >!; Working and/or reasoning: Correct Response: Question 6 The probability density function of a continuous random variable is given by: f x = 0.1 x 2 $ x 2 bx < x A 0 elsewhere (a) State the area bounded by the curve for 0 x 2 (b) The area bounded by the curve for 2 < x A must be: A B.?!" C. #!" D. 1 E Explain why C is the correct response (1 mark) 8

9 (c) (i) Explain why the graph of f(x) must be continuous at x=2 (ii) Hence show that the value of b = R >" (d) Find the value of A (1 + 2 = 3 marks) (e) Find the median of the distribution, correct to three decimal places. END OF PART 1 9