H2 Mathematics Probability ( )

H2 Mathematics Probability (208 209) Practice Questions. For events A and B it is given that P(A) 0.7, P(B) 0. and P(A B 0 )0.8. Find (i) P(A \ B 0 ), [2] (ii) P(A [ B), [2] (iii) P(B 0 A). [2] For a third event C, it is given that P(C) 0.5 and that A and C are independent. (iv) Find P(A 0 \ C). [2] (v) Hence state an inequality satisfied by P(A 0 \ B \ C). [] (200/P2/7) (i) P(A \ B 0 )P(B 0 ) P(A B 0 ) ( 0.) 0.8 0.32 (ii) P(A [ B) P(A \ B 0 )+P(B) 0.32 + 0. 0.92 (iii) P(B 0 A) P(A \ B0 ) P(A) 0.32 0.7 35 (iv) P(A 0 \ C) P(A 0 ) P(C) ( 0.7) 0.5 0.5 (since A and C are independent) (v) P(A 0 \ B \ C) apple P(A 0 \ C) ) P(A 0 \ B \ C) apple 0.5

2. The independent events A and B are such that P(A) x, P(B) x +0.5 and P(A [ B) 0.55, where x is a real constant. (i) Find the value of x. (ii) Find the probability that either A or B occurs, but not both. (iii) The event C is such that P(C B) 0.25. Find the probability that B occurs but C does not occur. (i) Since A and B are independent, P(A \ B) P(A) P(B) x (x +0.5) P(A [ B) P(A)+P(B) P(A \ B) ) 0.55 x +(x +0.5) x (x +0.5) ) x 2.85x +0.4 0 ) x 0.25 or. (rejected as 0 apple x apple 0.85) (ii) Probability P(A [ B) P(A \ B) 0.55 0. 0.45 (iii) P(C B) 0.25 P(B \ C) P(B) 0.25 P(B \ C) 0.25 0.4 0. Required Probability P(B) P(B \ C) 0.4 0. 0.3 2

3. Players A and B compete in a racquet match consisting of at most 3 sets. Each set is won by either Player A or B, and the match is won by the first person to win two sets. Player A has a probability of 2 3 of winning each of the first two sets. If the match goes into the third set, Player A has a probability of p of winning this set. (i) With the aid of a tree diagram, find in terms of p, the probability that player A will win the match. [3] (ii) Deduce the range of values of the probability that player A wins the match. [2] (iii) Show that the value of p in order for the match to be fair is 8. [] (iv) Given that Player A wins the match, find, in terms of p, the probability that he won the second set. [3] (i) P(A wins) 2 3 2 3 + 2 3 3 p + 3 2 3 p 4 9 (p + ) (NYJC/Prelim/20/P2/9) (ii) 0 apple p apple apple p +apple 2 4 9 apple 4 9 (p + ) apple 8 9 ) 4 9 apple P(A wins) apple 8 9 (iii) For the match to be fair, P(A wins) 2 4 9 (p + ) 2 p + 9 8 p 8 (iv) P(A won second set A wins) P(A won second set \ A wins) P(A wins) 2 3 2 3 + 3 2 3 p (p + ) 4 9 4 9 + 2 9 p 4 9 p + 4 9 2+p 2+2p 3

4. A company buys p% of its electronic components from supplier A and the remaining (00 p)% from supplier B. The probability that a randomly chosen component supplied by A is faulty is 0.05. The probability that a randomly chosen component supplied by B is faulty is 0.03. (i) Given that p 25, find the probability that a randomly chosen component is faulty. [2] (ii) For a general value of p, the probability that a randomly chosen component that is faulty was supplied by A is denoted by f(p). Show that f(p) 0.05p. Prove by di erentiation that 0.02p +3 f is an increasing function for 0 apple p apple 00, and explain what this statement means in the context of the question. [] p (i) P(faulty) 00 0.05 0.02p +3 00 p 25 ) P(faulty) 0.035 00 + p 00 0.03 (2009/P2/7) (ii) f(p) P(suppliedbyA faulty) P(supplied by A \ faulty) P(faulty) p 00 0.05 0.02p +3 00 0.05p 0.02p +3 f 0 0.05(0.02p + 3) (p) 0.02(0.05p) (0.02p + 3) 2 0.5 (0.02p + 3) 2 (0.02p + 3) 2 > 0 for 0 apple p apple 00 ) f 0 (p) > 0 and hence f is an increasing function The higher the percentage of components bought from supplier A, the higher the probability that a randomly chosen component that is faulty was supplied by A. 4

5. A bag contains 0 orange-flavoured, 4 strawberry-flavoured and cherry-flavoured sweets which are of identical shapes and sizes. Benny selects a sweet at random from the bag. If it is not cherry-flavoured he replaces it and selects another sweet at random. He repeats the process until he obtains a cherry-flavoured sweet. Calculate the probability that (i) the first sweet selected is strawberry-flavoured and the fourth sweet is orange-flavoured, [2] (ii) he selects an even number of sweets. [3] (i) Probability 4 24 24 0 3 2000 (HCI/Prelim/2007/P2/) (ii) Probability 24 a r 24 24 2 3 8 + 24 3 + 24 (sum to infinity of a GP) 5 + 24 7 +... 5

. There are 3 people at a BBQ party of two families. 25 people have the surname Tan and people have the surname Lim. The 25 people with the surname Tan consist of 4 single men, 5 single women and 8 married couples. The people with the surname Lim consist of 2 single men, 3 single women and 3 married couples. Two people are chosen at random from the party. (i) Show that the probability that they both have the surname Tan is 0 2. (ii) Find the probability that they are married to each other. (iii) Find the probability that they both have the surname Tan, given that they are married to each other. (iv) Find the probability that they are a man and a woman with the same surname. (v) Find the probability that they are married to each other, given that they are a man and a woman with the same surname. (i) Probability 25 C 2 3 C 2 0 2 (ii) Probability C 3 C 2 30 (there are married couples in total) (iii) Probability 8 (8 out of of the married couples have the surname Tan) (iv) Probability 2 C 3 C + 5 C C 3 C 2 3 05 (v) P(married same surname) P(married \ same surname) P(same surname) 30 3 05 8

7. An unbiased die is thrown times. Calculate the probabilities that the scores obtained will (i) consist of exactly two s and four odd numbers, (ii) be, 2, 3, 4, 5, in some order, (iii) have a product which is an even number, (iv) be such that a occurs only on the last throw and that exactly three of the first five throws result in odd numbers. (i) Probability 2 2 2 2! 2! 4! 5 92 (ii) Probability! 5 324 (iii) P(product is even) P(at least one of the scores is even) P(all scores are odd) 2 3 4 (iv) Probability 2 2 2 2 2 5! 3! 2! 5 2 7

8. In a particular country, bottles of cultured milk are manufactured by two companies, Yacoat and Vitergent. 0% of all cultured milk in that country is manufactured by Yacoat while the remaining % is manufactured by Vitergent. Both companies manufacture the cultured milk in three di erent flavours, Apple, Grape and Orange, and the percentage of each flavour manufactured by each of the two companies is given in the table below, where p and q are real constants. A bottle of cultured milk is randomly selected. (a) Find the probability that Apple Grape Orange Yacoat 50% 30% 20% Vitergent 00p% 0% 00q% (i) it is grape-flavoured, [2] (ii) it was manufactured by Yacoat, given that it is grape-flavoured. [] (b) Suppose that the probability that it is apple-flavoured and is manufactured by Vitergent is 0. less than the probability that it is orange-flavoured and manufactured by Vitergent. By setting up two equations involving p and q, find the values of p and q. [3] Three bottles are selected at random. (c) Find the probability that exactly one of them is grape-flavoured. (NJC/Prelim/205/P2/0 modified) (a) (i) Let E be the event that a randomly selected bottle of cultured milk is grape-flavoured. P(E) (0. 0.3) + (0.4 0.) 0.22 (ii) Let F be the event that a bottle of cultured milk is manufactured by Yacoat. P(F \ E) P(F E) P(E) 0. 0.3 0.22 9 or 0.88 (b) 0.4q 0.4p 0. ) q p 0.25 () p + q +0. ) p + q 0.9 (2) Solving () and (2): p 0.325, q 0.575 (c) Probability 0.22 0.78 0.78 3! 2! 0.544 8

9. An unbiased six-sided die is thrown multiple times. (i) Find the probability that at least one appears after the die is thrown exactly 5 times. (ii) Find, in terms of n, the probability that at least one appears after the die is thrown exactly n times. (iii) How many times must the die be thrown so that the probability of obtaining at least one is at least 0.99? Suppose now that the die is thrown until a appears. Find, in terms of n, the probability that (iv) it will take exactly n throws, (v) it will take at least n throws. Given that no appears in the first n throws, state the probability that a will appear in the the (n + ) th throw. (i) P(at least one ) P(no ) 5 45 777 5 (ii) P(at least one ) P(no ) 5 n (iii) 5 lg 5 n 5 0.99 n apple 0.0 n apple lg 0.0 lg 0.0 n lg 5 n 25.3 Hence the die must be thrown 2 times. (iv) Probability 5 n (v) Probability 5 a r n + 5 5 n 5 5 n n + 5 (sum to infinity of a GP) n+ + 5 n+2 +... Each throw of the die is independent, hence the probability of a appearing remains. 9

Further Practice Questions. In a popular weekly number game, a computer generates a combination of di erent numbers that can be any number from to 5. A player places a stake for a choice of di erent numbers from to 5. He wins the game if he gets at least four of the numbers correct. Let X be the number of correct answers he has chosen. The distribution of X is illustrated in the table below. x 0 2 3 4 5 2 08 P(X x) 75 75 Copy and complete the probability distribution table of X. (Give all the probabilities in terms of fractions in their lowest terms.) Hence find the probability of winning the game. Given that a player has won the game, what is the probability that he wins with five numbers? Note: P(X 0) 9 C 5 C 2 75 (given) and P(X ) 9 C 5 C 5 C 08 75 (given) P(X 2) P(X 3) P(X 4) P(X 5) P(X ) 9 C 4 C 2 5 C 54 43 9 C 3 C 3 5 C 48 43 9 C 2 C 4 5 C 08 00 9 C C 5 5 C 54 5005 C 5 C 5005 P(player wins) 08 00 + 54 5005 + 5005 7 43 P(X 5 player wins) P(X 5 \ player wins) P(player wins) 54 5005 7 43 54 595 0

2. Events A and B are such that P(B) 3,P(A \ B) 5 and P(A0 \ B 0 ). Find (i) P(A [ B), [] (ii) P(A B 0 ). [3] Hence determine whether events A and B are independent. [2] (i) P(A [ B) 5 (TPJC/Prelim/203/P2/5) (ii) P(A B 0 ) P(A \ B0 ) P(B 0 ) 5 3 4 3 3 P(A) P(A \ B 0 )+P(A \ B) 5 7 0 3 + 5 P(A) P(B) 7 30 P(A \ B) Hence A and B are not independent.

3. A teacher conducted a survey on a large number of students to determine the choice of colours for painting the school hall from 3 colour options of white, green and blue. Of the students surveyed, % were boys and 0% were girls. Of the boys, 50% chose white, 20% chose green and the rest chose blue. Of the girls, 25% chose white, 45% chose green and the rest chose blue. Draw a probability tree diagram to illustrate the above information. [] (i) One student is randomly selected. Find the probability that the student chose white. [] (ii) Two students are randomly selected. Find the probability that the two students are of the same gender or chose di erent colours (or both). [3] (iii) Three girls are randomly selected. Find the probability that exactly girl chose white, given that none of them chose blue. [3] (SAJC/Prelim/203/P2/7) (i) P(student chose white) (0.4 0.5) + (0. 0.25) 0.35 (ii) Required probability P(students are of di erent gender and chose same colour) 2[(0.4 0.5 0. 0.25) + (0.4 0.2 0. 0.45) + (0.4 0.3 0. 0.3)] 0.853 (iii) 0.443 2

4. The medical test for a certain infection is not completely reliable; if an individual has the infection there is a probability of 0.95 that the test will prove positive, and if an individual does not have the infection there is a probability of 0. that the test will prove positive. In a certain population, the probability that an individual chosen at random will have the infection is p. (i) An individual is chosen at random and tested. Show that the probability of the test being positive is 0. + 0.85p. (ii) Express in terms of p the conditional probability that a randomly chosen individual whose test is positive has the infection. Given that this probability is 0., find the conditional probability that a randomly chosen individual whose test is negative does not have the infection. (i) P(positive) p(0.95) + ( p)(0.) 0.+ 0.85p P(infected \ positive) (ii) P(infected positive) P(positive) 0.95p 0.+0.85p 95p 0 + 85p 95p 3 0. ) p 0 + 85p 22 P(not infected \ negative) P(not infected negative) P(negative) ( p)(0.9) (0.+0.85p) 0.99 3

5. A player throws three darts at a target. The probability that he is successful in hitting the target with his first throw is 8. For each of his second and third throws, the probability of success is twice the probability of success on the preceding throw if that throw was successful, the same as the probability of success on the preceding throw if that throw was unsuccessful. Construct a probability tree showing this information. [3] Find (i) the probability that all three throws are successful, [2] (ii) the probability that at least two throws are successful, [2] (iii) the probability that the third throw is successful given that exactly two of the three throws are successful. [4] (2007/P2/0) (i) P(SSS) 8 4 2 4 (ii) Probability P(SSS) + P(SSF) + P(SFS) + P(FSS) 4 + 8 4 + 2 8 3 4 4 2 25 + 7 8 8 4 (iii) P(3rd throw successful exactly two throws successful) P(3rd throw successful \ exactly two throws successful) P(exactly two throws successful) P(SFS) + P(FSS) P(SSF) + P(SFS) + P(FSS) 3 7 4

. A box contains 5 black balls and 3 red balls. 2 balls are drawn randomly from the box, one at a time. The colours of the balls that are drawn are noted. Let R and R 2 be the events that the first and second balls are red respectively. (a) Given that the draws are done with replacement, show that R and R 2 are independent events. (b) Given that the draws are done without replacement, find the probability of R 2. Hence show that R and R 2 are not independent events. (a) P(R \ R 2 ) 3 8 3 8 P(R 2 ) 5 8 3 8 + 3 8 3 8 3 8 P(R ) P(R 2 ) 3 8 3 8 P(R \ R 2 ) Therefore R and R 2 are independent events. (b) P(R 2 ) 5 8 3 7 + 3 8 2 7 3 8 P(R \ R 2 ) 3 8 2 7 P(R ) P(R 2 ) 3 8 3 8 P(R \ R 2 ) Therefore R and R 2 are not independent events. 5

7. A committee of 0 people is chosen at random from a group consisting of 8 women and 2 men. The number of women on the committee is denoted by R. (i) Find the probability that R 4. [3] (ii) The most probable number of women on the committee is denoted by r. By using the fact that P(R r) > P(R r + ), show that r satisfies the inequality (r + )! (7 r)! (9 r)! (r + 3)! >r! (8 r)! (0 r)! (r + 2)! and use this inequality to find the value of r. [5] (20/P2/) (i) P(R 4) 8 4 2 30 0 0.094 (ii) P(R r) > P(R r + ) ) 8 r 2 0 r 30 0 > 8 r+ 2 0 (r+) 30 0 8 2 8 2 > r 0 r r + 9 r 8! r! (8 r)! 2! (0 r)! (2 (0 r))! > 8! (r + )! (8 (r + ))! 2! (9 r)! (2 (9 r))! ) r! (8 r)! (0 r)! (2 + r)! > (r + )! (7 r)! (9 r)! (3 + r)! ) (r + )! (7 r)! (9 r)! (r + 3)! >r! (8 r)! (0 r)! (r + 2)! (r + )(r + 3) > (8 r)(0 r) r 2 +4r +3>r 2 28r + 80 32r >77 r>5.53 ) r

8. Given that events A and B are independent, show that A 0 and B are independent events. Given that P(A \ B) P(A) P(B), show that P(A 0 \ B) P(A 0 ) P(B). P(A 0 \ B) P(B) P(A \ B) P(B) [P(A) P(B)] P(A 0 ) P(B) [ P(A)] P(B) P(B) [P(A) P(B)] P(A 0 \ B) Therefore A 0 and B are independent events. 7