A purse contains 30 coins in total. Of these, 21 are one rupee coins and the remaining are 50 paise coins. Eleven coins are picked at random without replacement and placed in a box. If one coin is now picked at random from this box, what is the probability that it is a one rupee coin?

Introduction / Context:
This question involves a two stage random selection: first some coins are drawn from a purse into a box, and then a single coin is selected from the box. We need the probability that this final coin is a one rupee coin. The situation illustrates how symmetry can simplify probability computations that might otherwise look complicated.

Given Data / Assumptions:

• Total coins in purse = 30.
• One rupee coins in purse = 21.
• 50 paise coins in purse = 9.
• Eleven coins are drawn at random without replacement to fill the box.
• Then one coin is drawn at random from those 11 coins in the box.
• All coins are equally likely to be drawn at each stage.

Concept / Approach:
Although the description is two stage, each coin from the original 30 has the same overall chance of being the final selected coin. This is because the first sample of 11 and the final draw of 1 are both random and unbiased. Therefore, the probability that the final coin is a one rupee coin is the same as the fraction of one rupee coins in the original purse, which is much easier to compute than handling all combinations explicitly.

Step-by-Step Solution:
Total coins = 30.Number of one rupee coins = 21.We draw 11 coins into the box, then 1 coin from the box.Each coin in the purse has equal probability of 1 / 30 of being the final selected coin.This symmetry means that the chance the final coin is a one rupee coin is simply 21 / 30.Compute 21 / 30 = 7 / 10.Therefore, the required probability is 7 / 10.

Verification / Alternative check:
We could, in principle, count how many of the 11 coins in the box are one rupee coins using the hypergeometric distribution and then multiply by the conditional probability of drawing one of those one rupee coins. However, carrying out all these steps eventually simplifies to the same ratio 21 / 30. Recognizing the symmetry that any coin is equally likely to be the final draw saves a lot of computation and confirms our answer.

Why Other Options Are Wrong:
The options 4/7, 2/3 and 1/2 come from incorrect reasoning, such as assuming that the proportion of one rupee coins changes in a biased way when drawing the initial sample of 11. None of these fractions equals 21 / 30 after simplification, so they cannot be correct when every coin is equally likely to be the final selection.

Common Pitfalls:
Students sometimes attempt to track the exact distribution of one rupee coins in the box by conditioning on how many such coins were picked in the first draw, which leads to a long and error prone calculation. Another pitfall is thinking that the probability should depend on the intermediate sample size in a complicated way. Remember that when both stages are unbiased, the final probability of drawing a particular type of coin simply equals its original proportion in the population.

Final Answer:
The probability that the final selected coin is a one rupee coin is 7/10.