Why is it when the more possibilities there are, the less likely it is for each possibility to occur?

Question

I know how incredibly stupid this sounds. But something about this just doesn't click with me. Can someone please explain to me why when the more possibilities there are, the less likely for each possibility to occur? I get that it's because if the total sum of possibilities is 1, then the more possibilities there are, the more division to that "1" you are making. But that still doesn't really satisfy me. Can someone please help?

Answer 1

Think of a coin as a two-sided die. If you flip a coin, it is fairly likely (50%) that you get the outcome you want (say, heads).

Now if you are rolling a six-sided die and hoping for a six, you have to be a bit more lucky to roll a six on your first try.

Now imagine rolling a one-million sided die and hoping to land on $476,199$. You have to be incredibly lucky, because there are so many possible outcomes, and only one that you want.

Maybe a better example to illustrate the math: imagine throwing a dart at a square of area $1$. If you break it up into six parts, you have to hit the $1/6$ are of winning square. If you break it up into a million parts, you have to hit the $1/1000000$ area if winning square.

Answer 2

If the are $n$ options, the option with the greatest chance of success must have a probability of at least $\frac1n$. As $n$ increases, $\frac1n$ decreases, and so the lower bound for the greatest probability drops significantly.