Question

Problem 3. A chocolate box contains 10 large chocolate bars and 20 small chocolate bars at the beginning. Each day Bob chooses a chocolate bar from the box at random and eats it. What is the expected number of small chocolate bars remaining in the box when the first large chocolate bar is selected? (Hint: There are different methods for this problem. For example, you can first label the small chocolate bars by (1), (2), ..., (20) and define the indicator random variable X; = 1 if the (i) small chocolate bar is selected before the first large chocolate bar and X; = 0 otherwise. Then the number of small chocolate bars remaining when the first large chocolate bar is selected is 20 – X; and the question is to ask E[20 – E, X;].) [8 marks]

Answer 1

Answer: Given that choclate box contains 20 10 Small Choclate bars large choclate bars Now at to solve the first label the problem we use the given hirt small choclate bar as 1,2---20. Define, xi = riif ith small choclate hoor's selected before first longe choclate bar lo on prorth Small choclate boor is selected before ist large choclate box] 20 303 since L No. of small choclate 220 - Total no. of choclate= (2010)-30) (2010) ixi:r wp 213 20 OW : & P(xi= 1) = 2 Elxi)= 1PCXi =1) +0.p (Xi =6) - P(xi-1) = 2/3

Now 20 xi e No. of - belle 2. 1 small chocale has selected first loy chodarte har .. Perroining Sorcall chodate box: Cor? :. £xpected No. of crescaining storell Chocale best = {(202v1] Thole-fore, [[20-Exi] 20 - 20 20-E EX 720- PLY 9=1) E20- 420 - 60-40. 1. Expected no of remaining small choclate - 20 (answer)