Population, sample, and probability.

Say that 5% of a certain nation have above average IQ. Now, if I randomly pick 6 people, then what is the probability that one of them has above average IQ (i.e. is one of the 5%)?

I stumbled upon this question while reading a newspaper article some time ago. The writer gave the answer (26%) but I wasn't able to see why! And it kept haunting me. The last time I did any probability course was 15 years ago. So I hope someone would walk me through the logic of how to figure it out. Is there something missing here (that the writer used and didn't bother to spell out)?


Update: I still don't know the answer, but after playing with numbers I realised that the formula (1-0.95^n) gives the correct answers that I saw in the newspaper article. For example (1-.95^6 ~ 0.26). However, I'm not sure why this formula works and how it came about! Any comments and clarifications would be great.

To really understand a problem like this, I'd begin by breaking it down. The wording of the problem is ambiguous to me, but based on the answer the newspaper article gave and the formula you used, I believe I know what the problem is asking. The most general form of the problem is: Suppose you have a sample of n people, what is the odds of at least one person having above average IQ? The "at least" is important, because the odds of exactly one person having above average IQ is going to be different.

In any case, let's think about it logically. If the sample size is one person, then the odds is, obviously 5%. But, what if the sample size is more than one? Since we want to know the odds of at least 1 person being above average, it's easiest to look at the possible scenarios which don't meet our criteria. The only way for nobody to be above average is if everybody is average (the problem must assume for simplicity's sake that no one is ever below average). The odds of everyone being average are (95% * 95% * 95% ...) with 95% being multiplied n times. This can also be written as (0.95^n).

That odds we calculated are the odds of the event not happening, and we know that (the odds of the event happening) + (the odds of the event not happening) = 100%. We can then say that the odds of the event happening are 100% - (the odds of the event not happening). And since 100% can also be written as 1, we arrive at the formula you found. Does that help you make sense of the process?