ISS PREVIOUS YEAR 2016 PAPER-1 SET-A Q.no.- 11

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

ISS Previous year question 2016

Expected Number of Empty Boxes (ISS Level Problem)

Question 11

10 balls are placed in 10 boxes independently and uniformly at random. Initially, all boxes are empty.

What is the expected number of boxes that remain empty?

Options:

(a) (9/10)^9(b) 9^9 / 10^10(c) 9^10 / 10^9(d) (9/10)^10

Solution

This is a classic occupancy problem.

Step 1: Define Indicator Variables

Let Xi be defined as:

Xi = 1, if the i-th box is emptyXi = 0, otherwise

Then total number of empty boxes:

X = X1 + X2 + ... + X10

Step 2: Linearity of Expectation

E(X) = E(X1 + X2 + ... + X10)

Using linearity:

E(X) = E(X1) + E(X2) + ... + E(X10)

Since all boxes are identical:

E(X) = 10 × E(X1)

Step 3: Probability that a Box is Empty

For a given box:

Probability that a ball goes into that box = 1/10

Probability that a ball does NOT go into that box = 9/10

Since all 10 balls are independent:

P(box is empty) = (9/10)^10

Thus,

E(X1) = (9/10)^10

Step 4: Final Expectation

E(X) = 10 × (9/10)^10

Final Answer

E(X) = 10 (9/10)^10

Important Insights

• Indicator variable method is the most powerful approach

• Even if variables are dependent, expectation works due to linearity

• No need to calculate joint probabilities

Shortcut Trick (Very Important for MCQ)

For n balls and n boxes:

E(Number of empty boxes) = n (1 − 1/n)^n

For n = 10:

E(X) = 10 (9/10)^10

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