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 after ten tosses 12 1 0 5130E( ... ) 10E( ) $56.37 (nearest cent)91TT T T   

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 3 of 16 3 CJC Prelim 9233/2006/04/Q23(modified) 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). (i) 4 1 2 1 3 2 4 3)3(P             X (ii) Let X be the number of tries John takes to open the door. x 1 2 3 4 P(X = x) 4 1 4 1 3 1 4 3  1 4 4 1 2 1 3 2 4 3  (iii) By symmetry, 5E( ) 2X  5.74 144 134 124 1)(E 2222             X   222 5Var( ) E E 7.5 (2.5) 4XX X    E( 2) E( ) 2XX  = 5 2 + 2 = 9 2   5Var 2 Var 4XX 

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 4 of 16 4 TJC Prelim 9233/2005/02/Q12(modified) 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] Let X be the number of red balls drawn. Then X takes values 0, 1, 2. P0 P(fair die shows 1 or 2 and 2 black balls are drawn from urn A) +P(fair die shows 3, 4, 5 or 6 and 1 black ball is drawn from each of urns A and B) 154 254 5 5= ( s h o w n )3 8 7 3 8 8 168 X     132 243 9P2 387 388 5 6X        &    43P1 1 P0 P2 84XX X     x 0 1 2  P Xx 55 168 43 84 9 56 Hence,  55 43 9 5E0 12 168 84 56 6X      22 2 2 55 43 9 97E0 12 168 84 56 84X      2 22 97 5 29Var E E 84 6 63XX X     

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 5 of 16 5 RVHS Prelim 9758/2018/02/Q5 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] (i) Using the following possibility table, we have: 2nd die 1 2 3 4 5 6 1 1 1 1 1 1 1 2 1 2 2 2 2 2 1st die 3 1 2 3 3 3 3 4 1 2 3 4 4 4 5 1 2 3 4 5 5 6 1 2 3 4 5 6 From the table,  91P2 36 4X  Probability distribution of X: x 1 2 3 4 5 6 P Xx 11 36 9 36 7 36 5 36 3 36 1 36 A (ii)  6 1 91E( ) P (or 2.53) 36x Xx X x     22Var( ) E( ) E( )XX X

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________________ Additional Practice Questions for Chapter S2A: Discrete Random Variables Page 6 of 16   266 2 11 2 PP 301 91 36 36 2555 (or 1.97)1296 xx x Xx xXx               (iii) P(Ivan wins) 24 13 13 1 ...44 44 4          2 1 44 731 4  