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