RI S2A Discrete Random Variables Add Prac Qns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 1 of 5 Additional Practice Questions for Chapter S2A: Discrete Random Variables 1 A bag contains five red and two black counters. (a) Mary draws four counters from the bag, one by one at random and without replacement. The random variable X is used to denote the number of red counters drawn by Mary. Show that P( 22) 7X  and find the expectation of X. [4] (b) Peter pays $1 each time to draw a count er at random from the bag, one by one and with replacement until a black counter is obtained. Find the amount of money he should receive when a black c ounter is drawn in order for the game to be fair. [3] [You may assume 23 2 11 2 3 4 ... for | | 1.] (1 )xx x x x      2 A biased cubical die has faces numbered 1, 2, 3, 4, 5, 6 and X denotes the number obtained when the die is tossed once with probabilities 2P( )Xx k x . Find the value of k. [2] Joyce tosses the die and notes the number x obtained. At each toss, she will be given $(3x) if 1, 2, 3x  , and $( x) if 4, 5, 6x  . Calculate the expected amount Joyce will receive after ten tosses. [4] 3 John is given 4 keys of which only one can unlock the door. He tries the keys randomly (without replacement) to unlock the door. The random variable X denotes the number of tries John takes to unlock the door. (i) Show that P(X = 3) = 1 4 . (ii) Tabulate the probability distribution of X. (iii) Write down E(X) and find Var (X). Hence find E(X + 2) and Var (X + 2).

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 2 of 5 4 Urn A contains 5 black balls and 3 red balls. Urn B contains 4 black balls and 4 red balls. An experiment is conducted in the following manner: A fair die is tossed. If it shows 1 or 2, two balls are drawn from urn A without replacement. Otherwise, one ball is drawn from each urn. Let X be the number of red balls drawn in one experiment. Show that 55P( 0) 168X  . [1] Find the probability distribution of X. Find the exact values of expectation and variance of X. [6] 5 Two fair six-sided dice are thrown. The random variable X is the smaller of the two scores if they are different, and their common value if they are the same. (i) Show that 1P( 2) 4X  and find the probability distribution of X. [2] (ii) Hence find E( )X and Var( )X . [2] A game is played with Ivan and Jon taking turns to throw the two dice each. Ivan throws the dice first and the player who first obtain the value of X equals to 2 wins the game. If Ivan wins the game, Jon pays him $7. Otherwise, Ivan pays Jon $10. (iii) Find the player who has the higher expected gain. Justify your answer. [3] 6 In a game with a 4-sided fair die numbered 1 to 4 on each face, the score for a throw is the number on the bottom face of the die. A player gets to choose either option A or option B. Option A: The player rolls the die once. The score x is the amount of money $x that the player wins. Option B: The player rolls the die twice. The first score is x and the second score is y. If y > x, the player wins $2xy, but if y < x, the player loses $(x – y). Otherwise, he neither wins nor loses any money. (i) Find the expected amounts won by a play er in one game when playing option A and when playing option B. Show that option B is a better option. [5] (ii) Suggest why a risk averse player would still choose option A. [1] (iii) Show that the variance of the amount w on by a player in one game when playing option A is $21.25. [2]

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 3 of 5 7 A particle moves one step each time either to the right or downwards through a network of connected paths as shown above. The particle starts at S, and, at each junction, randomly moves one step to the right with probability p, or one step downwards with probability q, where 1qp . The steps taken at each junction are independent. The particle finishes its journey at one of the 6 points labelled iA , where 0, 1, 2, 3, 4, 5i  (see diagram). Let   X i be the event that the particle arrives at point iA . (i) Show that   23P2 1 0X pq . [2] (ii) After experimenting, it is found that the particle will end up at point 2A most of the time. By considering the mode of X or otherwise, show that 11 32 p . [4] The above setup is a part of a two-stage computer game.  If the particle lands on 0A , the game ends immediately and the player will not win any points.  If the particle lands on iA , where i = 2 or 4, then the player gains 2 points.  If the particle lands on iA , where i = 1, 3 or 5, then the player proceeds on to the next stage, where there is a probability of 0.4 of winning the stage. If he wins the stage, he gains 5 points; otherwise he gains 3 points. Let Y be the number of points gained by the player when one game is played. (iii) If 0.4p  , determine the probability distribution of Y. [4] (iv) Hence find the expectation and variance of Y. [2]

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 4 of 5 8 The probability of obtaining a ‘6’ on a biased cubical dice is thrice the probability of rolling every other number on the dice. Show that probability of obtaining a ‘6’ on the biased dice is 3 8 . [1] This biased dice is put into a bag together with 3 fair cubical dice. (i) One of the dice is chosen randomly fr om the bag and rolled once. Find the probability of obtaining a ‘6’. [2] (ii) One of the dice is chosen randomly and is rolled n times. (a) Find the probability that ‘6’ is obtained on all the n rolls. [1] (b) Given that ‘6’ is obtained for all the n rolls, the probability that the biased dice is chosen is more than 0.95. By forming an inequality in terms of n, solve for the least value of n. [3] (iii) One of the dice is chosen randomly from th e bag and rolled once. The dice is then put back into the bag. Another dice is chosen randomly from the bag and rolled once. For every ‘6’ obtained, the player gains $2, otherwise the player loses $ k. If the game is fair, that is, expected winnings of the player is $0, show that the value of k is 0.56. [3] Find the variance of the player’s winnings. [2] 9 Shania and Tina are playing a game. Shania has a bag containing one green disc, r red discs and 2r blue discs, where r > 1. A red disc is worth 5 points, a blue disc is worth 2 points and the green disc is worth 0 points. Tina takes two discs from the bag at random. Tina’s score is found by multiplying together the number of points for each of the two discs she takes. (i) State Tina’s pos sible scores. [1] (ii) Show that the expectation of Tina’s score is 27 11 ,31 r r   and find an expression for the variance. [7] (i