# Topic 5 Review [81 marks]

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1 Topic 5 Review [81 marks] A four-sided die has three blue faces and one red face. The die is rolled. Let B be the event a blue face lands down, and R be the event a red face lands down. 1a. Write down (i) P(B); (ii) P(R). (i) P(B) N1 (ii) P(R) N1 = 4 = 1 4 1b. If the blue face lands down, the die is not rolled again. If the red face lands down, the die is rolled once again. This is represented by the following tree diagram, where p, s, t are probabilities. Find the value of p, of s and of t. p = 4 N1 s = 1, 4 4 N1 1c. Guiseppi plays a game where he rolls the die. If a blue face lands down, he scores 2 and is finished. If the red face lands down, he scores 1 and rolls one more time. Let X be the total score obtained. (i) Show that P(X = ) =. (ii) Find P(X = 2). [ marks]

2 (i) P(X = ) = P (getting 1 and 2) = 1 4 = AG N (ii) P(X = 2) = + (or 1 ) () = 1 [ marks] 1d. (i) Construct a probability distribution table for X. (ii) Calculate the expected value of X. (i) A2 (ii) evidence of using E(X) = xp(x = x) 1 E(X) = 2( ) + ( ) 5 = (= 2 ) () 1e. If the total score is, Guiseppi wins \$10. If the total score is 2, Guiseppi gets nothing. Guiseppi plays the game twice. Find the probability that he wins exactly \$10. win \$10 scores one time, 2 other time 1 P() P(2) = (seen anywhere) evidence of recognising there are different ways of winning \$10 e.g. P() P(2) + P(2) P(), 2( ), P(win \$10) = (= ) 1 N The ages of people attending a music concert are given in the table below. 2a. Find p.

3 evidence of valid approach e.g , line on graph at x = 1 p = 144 2b. The cumulative frequency diagram is given below. Use the diagram to estimate (i) (ii) the 80th percentile; the interquartile range. (i) evidence of valid approach e.g. line on graph, 0.8 0, using complement = 2.5 (ii) Q 1 = 2 ; = 2 ()() Q IQR = 6 (accept any notation that suggests an interval) N

4 The histogram below shows the time T seconds taken by children to solve a puzzle. The following is the frequency distribution for T. a. (i) Write down the value of p and of q. (ii) Write down the median class. [ marks] (i) p = 17, q = 11 (ii) 75 T < 85 N1 [ marks] b. A child is selected at random. Find the probability that the child takes less than 5 seconds to solve the puzzle. evidence of valid approach e.g. adding frequencies 76 = P(T < 5) = =

5 c. Consider the class interval 45 T < 55. (i) (ii) Write down the interval width. Write down the mid-interval value. (i) 10 N1 (ii) 50 N1 d. Hence find an estimate for the (i) (ii) mean; standard deviation. (i) evidence of approach using mid-interval values (may be seen in part (ii)) x = 7.1 A2 N (ii) σ =.4 N1 e. John assumes that T is normally distributed and uses this to estimate the probability that a child takes less than 5 seconds to solve the puzzle. Find John s estimate. e.g. standardizing, z = P(T < 5) = 0.8 There are nine books on a shelf. For each book, x is the number of pages, and y is the selling price in pounds ( ). Let r be the correlation coefficient. 4a. Write down the possible minimum and maximum values of r. min value of r is 1, max value of r is 1

6 4b. Given that r = 0.5, which of the following diagrams best represents the data. [1 mark] C N1 [1 mark] 4c. For the data in diagram D, which two of the following expressions describe the correlation between x and y? perfect, zero, linear, strong positive, strong negative, weak positive, weak negative linear, strong negative

7 Two boxes contain numbered cards as shown below. Two cards are drawn at random, one from each box. 5a. Copy and complete the table below to show all nine equally likely outcomes. A2 5b. Let S be the sum of the numbers on the two cards. Find the probability of each value of S. P(12) = 1, P(1) =, P(14) =, P(15) = 2 A2 5c. Find the expected value of S. [ marks] correct substitution into formula for E(X) 1 e.g. E(S) = E(S) = 12 [ marks] A2 2 5d. Anna plays a game where she wins \$50 if S is even and loses \$0 if S is odd. Anna plays the game 6 times. Find the amount she expects to have at the end of the 6 games. [ marks]

8 METHOD 1 correct expression for expected gain E(A) for 1 game e.g () E(A) = 50 amount at end = expected gain for 1 game 6 = 200 (dollars) METHOD 2 attempt to find expected number of wins and losses e.g , 6 attempt to find expected gain E(G) e.g E(G) = 200 (dollars) [ marks] Consider the events A and B, where P(A) = 0.5, P(B) = 0.7 and P(A B) = 0.. The Venn diagram below shows the events A and B, and the probabilities p, q and r. 6a. Write down the value of (i) p ; (ii) q ; (iii) r. [ marks] (i) p = 0.2 N1 (ii) q = 0.4 N1 (iii) r = 0.1 N1 [ marks] 6b. Find the value of P(A B ).

9 P(A B ) = 2 A2 Note: Award for an unfinished answer such as c. Hence, or otherwise, show that the events A and B are not independent. [1 mark] valid reason R1 2 e.g. 0.5, thus, A and B are not independent AG N0 [1 mark] 7a. A factory makes lamps. The probability that a lamp is defective is A random sample of 0 lamps is tested. Find the probability that there is at least one defective lamp in the sample. evidence of recognizing binomial (seen anywhere) e.g. B(n, p), finding P(X = 0) = () appropriate approach e.g. complement, summing probabilities probability is N 7b. A factory makes lamps. The probability that a lamp is defective is A random sample of 0 lamps is tested. Given that there is at least one defective lamp in the sample, find the probability that there are at most two defective lamps. identifying correct outcomes (seen anywhere) () e.g. P(X = 1) + P(X = 2), 1 or 2 defective, recognizing conditional probability (seen anywhere) e.g. P(A B), P(X 2 X 1), P(at most 2 at least 1) appropriate approach involving conditional probability P(X=1)+P(X=2) e.g.,, probability is P(X 1) or R1

10 A company produces a large number of water containers. Each container has two parts, a bottle and a cap. The bottles and caps are tested to check that they are not defective. A cap has a probability of of being defective. A random sample of 10 caps is selected for inspection. 8a. Find the probability that exactly one cap in the sample will be defective. Note: There may be slight differences in answers, depending on whether candidates use tables or GDCs, or their sf answers in subsequent parts. Do not penalise answers that are consistent with their working and check carefully for FT. evidence of recognizing binomial (seen anywhere in the question) e.g. nc r p r q n r, B(n, p), 10 C 1 (0.012) 1 (0.88) p = b. The sample of caps passes inspection if at most one cap is defective. Find the probability that the sample passes inspection. valid approach e.g. P(X 1), p = 0.4 8c. The heights of the bottles are normally distributed with a mean of 22 cm and a standard deviation of 0. cm. (i) Copy and complete the following diagram, shading the region representing where the heights are less than 22.6 cm. (ii) Find the probability that the height of a bottle is less than 22.6 cm.

11 (i) Note: Award for vertical line to right of mean, for shading to left of their vertical line. (ii) valid approach e.g. P(X < 22.6) working to find standardized value e.g., 2.1 p = 0.82 N () 8d. (i) A bottle is accepted if its height lies between 21.7 cm and 22.6 cm. Find the probability that a bottle selected at random is accepted. (ii) A sample of 10 bottles passes inspection if all of the bottles in the sample are accepted. Find the probability that the sample passes inspection. valid approach e.g. P(21.7 < X < 22.6), P( 2.1 < z < 2.1) correct working () e.g (1 0.82) p = 0.64 N (ii) correct working () e.g. X B(10,0.64), (0.64) 10 p = 0.65 (accept 0.64 from tables) 8e. The bottles and caps are manufactured separately. A sample of 10 bottles and a sample of 10 caps are randomly selected for testing. Find the probability that both samples pass inspection. valid approach e.g. P(A B) = P(A)P(B), (0.4) (0.64) 10 p = 0.61 (accept 0.60 from tables) International Baccalaureate Organization 2014 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for East Mecklenburg High School

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