RI 2025_Y6T2W4Mathfocus_DRV+and+Binomial+Distribution_Soln

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _________________________________________________________________ Y6 H2 Math Term 2 W4 Math Focus: Discrete Random Variables, Binomial Distribution Page 1 of 4 Term 2 W4 Math Focus: Discrete Random Variables, Binomial Distribution Questions 1 JPJC BT2 9758/2021/02/Q9 A biased four-sided die has the numbers ‘1’ to ‘4’ labelled on its faces. When the die is thrown, t he number shown on the die is the score, X , which follows the probability distribution: ( ) , 1, 2 P 11 , 3, 44 kx x Xx kxx == =  −= where k is a non-zero positive constant. (i) Show that the value of k is 1 6 . [2] In a game, Evan tosses two such dice. The probability distribution of Y, which is the sum of the scores shown on the dice, is given in the following table: y 2 3 4 5 6 7 8 ( )P Yy= a 1 9 5 24 b 43 192 35 288 25 576 (ii) Find the values of a and b. [3] (iii) If Y is an odd number, Evan wins $Y. If Y is an even number, Evan loses $Y. Let W be Evan’s winnings after one game. Find the expectation and variance of W. [3] Solution: (i) Total probability = 1 ( ) ( ) 11 1112 134 44 55 2 12 1 6 kk k k k k + +− +− = = = x 1 2 3 4 ( )P Xx= 1 6 1 3 7 24 5 24 (ii) Sum 1 (1/6) 2 (1/3) 3 (7/24) 4 (5/24) 1 (1/6) 2 3 4 5 2 (1/3) 3 4 5 6 3 (7/24) 4 5 6 7 4 (5/24) 5 6 7 8 Note: The outcomes are not equally likely to happen given that it is not a fair die.

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________ ____________________________________________________________________________ Y6 H2 Math Term 2 W4 Math Focus: Discrete Random Variables, Binomial Distribution Page 2 of 4 ( ) 11 1P2 6 6 36aY= = =×= ( ) 1 5 1 7 19P5 2 2 6 24 3 24 72bY= == ×× + ×× = Alternatively, ( ) ( ) 19P 5 1P 5 72bY Y= == − ≠= (iii) y 2 3 4 5 6 7 8 w -2 3 -4 5 -6 7 -8 ( )P Ww= 1 36 1 9 5 24 19 72 43 192 35 288 25 576 ( ) 1 19 35 1 5 43 25E ( ) 3 57 246 89 72 288 36 24 192 576 11 144 0.08 to 2 d.p. W =×+× +× −× −× −× −× =− =− 2 1 19 35 1 5 43 25E( ) 9 25 49 4 16 36 649 72 288 36 24 192 576 8017 288 W = × +×+× + ×+×+× +× = [ ] 2 22 8017 11Var( ) E( ) E( ) 27.8288 144WW W = − = −− =  Alternatively, use 1-Var stats: 2 SAJC BT2 8865/2021/10 A glass vial is an important primary packaging component for pharmaceutical drugs. Vaccines, and most injectable drugs, need to be packaged in these sterile glass vials. On average, it was found that 5% of glass vials shipped by a certain pharmaceutical company are defective. The glass vials are packed in boxes. Each box consists of 50 glass vials. (i) State, in context, two assumptions needed for the number of defective vials in a box to be well modelled by a binomial distribution. [2] Assume now that the number of defective vials in a box of shipment follows a binomial distribution.