2026 RVHS H2 J2 Revision Package (Probability,Vectors, Complex Numbers) - Questions

1 Probability 1 NYJC Prelim 9758/2023/02/Q6 In a funfair, a vendor sells lucky draw tickets at $1 per ticket. He accepts payment using $1 or $2 notes only. In the queue for tickets, there are m people each paying with a single $1 note and n people each paying with a single $2 note. Each person in the queue wants to buy a single ticket and each arrangement of people in the queue is equally likely to occur. Initially, the vendor has no money but a large supply of tickets. The vendor will stop selling tickets if he cannot give the required change. (a) In the case of 1m and 1n= , find the probability that the vendor can sell one ticket to each person in the queue. [2] (b) By considering the first three people in the queue in the case of 2m and 2n= , (i) show that the probability that the vendor is able to sell one ticket to each person in the queue is 1 1 m m − + , [3] (ii) find the probability that the vendor is unable to sell any ticket to the third person, given that the vendor is unable to sell tickets to the first three people in the queue. [3] 2 PJC Prelim 9758/2017/02/Q10 For events A and B, it is given that 11P( ) 20A = and 1P( ) 2B = . (i) Find the greatest and least possible values of P( )AB . [2] It is given in addition that 7P( | ') 9BA = . (ii) Find P( )AB . [2] (iii) Determine if A and B are independent events. Justify your answer. [2] (iv) Given another event C such that 2P( ) 5C = , 19P( ) 20A B C  = , 1P( ) 10A B C  = and P( ) 2P( )A C B C =  , find P( )AC . [3] River Valley High School 2026 JC2 H2 Mathematics Revision Package for Lecture Test 1

2 3 MI Prelim 9758/2020/02/Q8 A group of n people each takes turn to pick one card, with replacement, from a pack of cards. (a) Assume that there are 12 unique cards in a pack. The probability that all the people in the group picked different cards is denoted by P. (i) For the case where 3n= , show that 55 72P= . [2] (ii) Find P for the case where 4n= and deduce the smallest value of n such that 1 2P . [2] (iii) State, with a reason, the least possible value of n such that 0P= . [2] (b) Assume now that there are 120 unique cards in the pack. It is given that, for the case where 12n= , the probability that all the people in the group picked different cards is 0.56640, correct to 5 decimal places. Find the smallest value of n such that the probability that at least two people picked the same card exceeds 1 2 . [4] 4 CJC MYE 9758/2021/Q10 A group of 11 friends, including Alan, Ben and Chris, forms a team to play football friendlies matches. In the team, there has to be 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. Everyone in the team can play in any position. (i) Find the number of different teams that the 11 friends can form.