RI S2A Discrete Random Variables Tutorial Qns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 __________________________________ Tutorial S2A: Discrete Random Variables Page 1 of 4 Tutorial S2A: Discrete Random Variables Section A (Discussion Questions) 1 Find the mean and standard deviation of the random variable X with the following probability distribution: x 4 3 2 1 0 1 2 3 P Xx 0.04 0.16 0.24 0.16 0.15 0.1 0.1 0.05 [ –0.83, 1.86] 2 Two dice are thrown and the numbers A and B shown on each die are noted. The score X from the throw is defined by: if , if . A BA B X A BA B    (i) Tabulate the probability distribution of X. (ii) Evaluate the exact value of E(X) and Var(X). [(i) 28 9 (ii) 1015 162 ] 3 A discrete random variable X takes values 2, 3, 4, 5 with probabilities as shown in the table. x 2 3 4 5 P( )Xx k 4 k 9 k 16 k (i) Find k, leaving your answer as a fraction. [2] (ii) Find E( cos )X , giving your answer to 3 significant figures. [3] (iii) Find 12P( 7)XX , where 1X and 2X are two independent observations of X. [3] [(i) 144 205k  (ii) 0.530 (iii) 0.0891]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ __________________________________ Tutorial S2A: Discrete Random Variables Page 2 of 4 4 A box contains 2 fair tetrahedral dice. The firs t die has sides labeled 1, 1, 2, and 3 and the second die has sides labeled 1, 2, 3 and 3. A die is taken at random from the box and thrown. X, the score is defined as follows: If the first die is picked, then it is thrown and the score is defined as two times the number which appears on the base of the first die. If the second die is picked, then it is thrown and the score is the number which appears on the base of the second die. Show that P(X = 2) = 3 8 , and find the probability distribution of X. (i) Show that E(X) = 23 8 and find the exact value of Var(X). (ii) If this experiment is performed twice, find the probability that the score is 3 for the first experiment given that the total score is 4. (iii) Tom and Jerry take turns to perform this expe riment with Jerry playing first. They stop when one of them obtains a score of 2. What is the probability of Tom obtaining a score of 2? [(i) 135 64 (ii) 2 13 (iii) 5 13 ] 5 Alfred and Bertie play a game, each starting with cash amounting to £100. Two dice are thrown. If the total score is 5 or more then Alfred pays £x, where 08 x , to Bertie. If the total score is 4 or less, then Bertie pays £(x + 8) to Alfred. By finding the probability distribution of Y, where Y denotes the random variable representing Alfred’s gains after one game, show that the expectation of Alfred’s cash after the first game is £ 1 3 (304 – 2x). Find the expectation of Alfred’s cash after six games. Find the value of x for the game to be fair. [Hint: For the game to be fair, expectation of Alfred’s gains needs to be equal to 0.] Given x = 3, find the variance of Alfred’s cash after the first game. [£(108 – 4x), x = 2, £2 245 9 ]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ __________________________________ Tutorial S2A: Discrete Random Variables Page 3 of 4 6 A circular card is divided into 3 sectors scoring 0, 1, 2 and having angles 135 ,135 and 90 respectively. The card is mounted onto a wall at its centre such that it can rotate freely. A pointer is fixed on the wall just above the card, as shown in the diagram. In a game, a participant will spin the card twice, and the random variable X is the product of the scores of the 2 spins. (i) Tabulate the probabi lity distribution of X. [3] (ii) Show that  49E 64X  , and find Var X . [2] Dave intends to use the above setup in a fundraising carnival. For every game, the participant will be charged $h to play, and will be awarded a prize money of $5 X based on the outcome of their spins. Find the least integer value of h such that on average, Dave can expect to earn at least $1 from each participant. [2] [(ii) 5343 4096 , 5] 7 A bag contains (5 )n numbered balls. Two of the balls are numbered 3, three of the balls are numbered 4 and n of the halls are numbered 5. Two ba lls are taken, at random and without replacement, from the bag. The random variable S is the sum of the numbers on the two balls taken. (i) Determine the probability distribution of S. [4] (ii) For the case where n = 1. find P(S = 10) and explain this result. [1] (iii) Show that E( S) = 10 36 5 n n   and Var(S) =  2 g( ) 5( 4 ) n nn where g( )n is a quadratic polynomial to be determined. [6] [(ii) 0] 1 2

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ __________________________________ Tutorial S2A: Discrete Random Variables Page 4 of 4 8 Tom has a bag of wooden rectangular blocks of identical size. The bag contains 1 blue block, m red blocks and  1m yellow blocks, where 2m  . Tom and Jerry play a game using Tom’s bag of wooden blocks. Jerry draws 2 bl ocks at random, one at a time, without replacement. 3 points will be awarded if a yellow block is drawn, 2 points will be awarded if a red block is drawn, but no points will be awarded if a blue block is drawn. Jerry’s final score is the product of the points awarded for the 2 blocks drawn. Let the random variable X denotes Jerry’s final score. (i) Show that  1P0X m and hence, find the probability distribution of X. [4] (ii) Find the value of m if Jerry’s expected final score is 5. [2] Tom pays Jerry $5 if Jerry’s final score is at least 5 and Jerry pays Tom $a if his final score is less than 5. (iii) Using the value of m found in (ii), find the range of values of a if Tom is expected to make a profit. [2] [(ii) 6m ; (iii)  7.70 to 2dpa ] 9 A committee of 10 people is chosen at random from a group consisting of 18 women and 12 men. The number of women on the committee is denoted by R. (i) Find the probability that 4R . (ii) The most probable number of women on the committee is denoted by r. By using the fact that   PP 1Rr Rr   , show that r satisfies the inequality              1 ! 17 ! 9 ! 3 ! ! 18 ! 10 ! 2 !r r rr r r rr     and use this inequality to find the value of r. (iii) Use the calculator to find E R and Var R . [(i) 0.0941 (ii) 6 (iii) 6, 1.66] THE END