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# Fishing for Probability, Part I

October 28, 2016

So you want to explain probability to your child? Or maybe you want to understand it a bit more yourself? Consider this example. I've done it with a class of 10 year olds before and they seemed very receptive.

You fish in a very small lake. A real small one. There are 4 grey fish in the lake and 3 gold fish. Yes, just 7 fish in that lake. As stated, it is small. You reach the lake early in the morning and your plan is to fish non-stop until you come out with exactly 3 fish. That is your quota. The fish go for the bait easily, so you won't have to wait too long. In aiming to fish 3/7 (about 43%) of the lake's population, you can't claim to be a great conservationist. But I certainly forgive you. After all, without you, this example won't exist.

Being the nature killer that you are, say that you wish to catch as many gold fish as possible (up to 3). But unfortunately, the exact number of gold fish you catch isn't really at your control. Instead, it is up to a matter of chance to determine which fish go for your bait. What are the possibilities?

Well, you are going to come out with 3 fish - that is for sure. Then there are 4 possible options:

X = 0, you come out with 0 gold fish (unlucky you). All your fish are grey. At least these might be tasty.

X = 1, you come out with 1 gold fish (and 2 grey fish).

X = 2, you come out with 2 gold fish (and 1 grey fish).

X = 3, you population killer! You come out with all 3 gold fish. On a positive note, you left the quad of grey fish at peace.

Well, it is a matter of chance right? X is indeed a random quantity. What are then the chances of having X = 0, X = 1, X = 2 or X = 3?

You ask then: What is "chance"? I then say that it is a number between 0 and 1 quantifying the likelihood of the outcome. This means, that if for example the chance of X = 2, happens to be 0.34, then if you would live this life a million times and go to that lake in each life, then for about 340,000 of these lives you'd get X = 2 gold fish. Does that make sense? Sure it, does. The chance of you never thinking about probability before is almost 0.

So is it then the case that the chance of each of the outcomes is 0.25? After all, there are 4 possible outcomes to this day of fishing. Well... if you think about it for a minute you'll convince yourself that this isn't plausible:

(A) There are more grey fish than gold ones. Hence it is more likely to have all grey fish (X = 0) in comparison to all gold fish (X = 3).

(B) Using similar reasoning you might see that X = 1 (1 gold and 2 grey) is more likely than X = 2 (2 gold and 1 grey). Can you argue why?

Now here is the nice bit about this day of fishing. Try to explain this to your 9-15 year old daughter or son and ask them to assign four numbers (probabilities) for each of the possible outcomes of X. Their probabilities need to sum up to 1 of course. Assuming that you catch them (the kids I'm talking about here) at a good time, I'm quite certain that you'll be positively surprised. You'll see that they have a good intuition for this problem and they will come up with numbers that agree with (A) and (B) above.

Want to know the correct exact probabilities? What do I mean by that? Well, unless you've thought about the hypergeometric distribution before, you may wish to continue to Fishing for Probability, Part II.

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