ISS PREVIOUS YEAR 2016 PAPER-1 SET-A Q.no.- 10

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

ISS Previous year question 2016

Expected Number of Unique Elements in a Sample

Question

A simple random sample of size 10 is selected with replacement from a population of size 100. What is the expected number of unique elements in the sample?

Options:(a) 100 × (99/100)¹⁰(b) 100 × [1 − (99/100)¹⁰](c) 100 × (9/10)¹⁰(d) 100 × (1 − 99/100)¹⁰

Solution

This problem is based on the concept of expected number of distinct (unique) elements.

Consider any one element from the population.

• The probability that this element is not selected in one draw is:

  
  

  99/100

• Since sampling is done with replacement, the draws are independent.

  
  

  So, the probability that the element is never selected in 10 draws is:

  
  

  (99/100)¹⁰

• Therefore, the probability that the element is selected at least once is:

  
  

  1 − (99/100)¹⁰

Now, there are 100 such elements in the population.

so, the expected number of elements that appear at least once in the sample is:

Expected number of unique elements =100 × [1 − (99/100)¹⁰]

Final Answer

Option (b):100 × [1 − (99/100)¹⁰]

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

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