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conantee
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« on: December 31, 2009, 07:20:07 AM »

15. A sample of N atoms of helium gas is confined in a 1.0 cubic meter volume. The probability that none of the helium atoms is in a 1.0 \times 10^{-6} cubic meter volume of the container is
(A) 0
(B)  (10^{-6})^N
(C) (1-10^{-6})^N
(D)  1-(10^{-6})^N
(E) 1
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conantee
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« Reply #1 on: December 31, 2009, 07:29:34 AM »

This is another example of how to use limit to distinguish answer.

First, (A) and (E) are wrong since they do not depend on N.

Next, as N is large, the probrability that none of the helium atoms in a certain volume is extremely low, since more atoms tends to spread over the place. So, (D) is wrong since as N is large, (D) goes to one.

Now, let N = 1. That is, consider only one atom. Now, the problem is easy. The probability that an atom will be in a certain volume is proportional to the volume itself. So, the probability that one atom will not be in a 1.0 \times 10^{-6} cubic meter volume is then 1-\frac{1.0 \times 10^{-6}}{1} = 1- 10^{-6}

This makes (B) wrong. The only answer left is (C).

The correct answer is (C), and 36 out of 100 people got the right answer.
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