RI S2A Discrete Random Variables_Add Prac_Solns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 1 of 16 Additional Practice Questions for Chapter S2A: Discrete Random Variables (Solutions) 1 JJC Prelim 9233/2005/02/Q12 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      (a) Let X be the number of red counters drawn. X can take values 2, 3 or 4. P(X = 2) = P(2 R & 2 B in any order) 5421 4 ! 7 6 5 4 2!2! 2 [Shown]7 P(X = 3) = P(3 R & 1B in any order) 54324 ! 4 =76543 ! 7 P(X =4) = P(4 R) 5432 1 =7654 7 241 2 0E ( ) 234 777 7X  (b) For fair game, E(amt. received) = E(amt. paid) 23 25 2 5 2 5 2$1 $2 $3 $4 ....77 7 7 7 7 7         23 25 5 5$ [1 2 3 4 ....]77 7 7          2 21$ 57 (1 ) 7       $3.50

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 2 of 16 2 NYJC Prelim 9233/2005/02/Q12 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] 6 2 1 119 1 1 91x kx k k    Let T be the amount she gets after one toss. t 3 6 9 4 5 x 1 2 or 6 3 4 5 P( )Tt 1 91 2226 4 0 91 91 91 9 91 16 91 25 91 1 40 9 16 25 513E( ) 3 6 9 4 591 91 91 91 91 91T  Hence expected amount she gets a