I loved probability theory both at school and at uni, where the theory was applied to studying populations of tagged/untagged specimens in the wild.

The answers are correct - we often get bamboozled by probability, thinking that everything in the world is connected far more than it is. We might sit at a poker machine (slot machine or whatever you call it) and pour in coin after coin and think, "I've got to get a payoff soon, it's got to be my turn soon." But in reality, you have no better chance with each coin than with all the ones before it. As Kathy said, this is random. We just like to hope that it's not. The classic, simple question is: "A kid tosses a coin and gets 99 heads in a row. What is the chance that the next toss will also be a head?"

The answer is, 1 in 2. Or, 0.5. However, in practical terms, I'd be checking the coin to make sure there is no bias, such as having two heads. If the coin DOES have two heads, then the next coin toss is a certainty to get a head.

And yes, to get the probability of consecutive events, you multiply them together, but sometimes you have to be slightly sneaky about it and multiply the probabilities of the events NOT happening, then subtract your final answer from 1.

The second example is an interesting one - it's "sampling without replacement". If you follow it through, the more you sample without replacement, the better your chances each time of getting it right. The safe in "The Price Is Right" is a good example - it has three knobs. Each knob has three numbers. You can't use the same number twice (which makes it sampling without replacement). The chance of getting the first knob right is 1 in 3. The chance of getting the second knob right is 1 in 2. The last knob has no choices left, but the remaining number, so if the first two are right, then this last number is automatically correct - probability 1. The total probability is one third times one half, or 1 in 6. The more numbers you have to choose from (and the more knobs), the process simply continues. You can see that for 10 knobs and 10 digits, you're multiplying 1 over 10 x 9 x 8 x 7 x 6 x 5..etc. This is more easily written as 10! or 10 factorial. Writing it as 10! (or whatever number) saves paper. So if you ever see that little ! after a number, that's what it means. It is NOT mathematicians becoming excited.

We use "sampling WITH replacement" whenever tagged animals are released into the wild. Have you ever wondered why scientists bother to tag animals? You can't tag 'em all, and how can you get useful information from just half a dozen tagged crocodiles, for example?

This was back in the days when animals were tagged without radio tracking - the tag had to be attached in a way that made recapture as likely as capture of a fresh, previously uncaptured specimen. Let's say we capture and tag 20 specimens, then release them. We lay nets again, and capture another 20. Of that 20, 5 wear tags (and are therefore recaptures). What is the total population in the wild?

It's actually not difficult, if you think about it logically. The group captured is (hopefully) a fully random representative sample. Of the ones we've just captured, 5 wear tags. This means we tagged 5 out of 20, or 20%, of the population. Therefore 20 specimens tagged first go, is 20% of the population. Total population - about 100.

You can't be exact because there are always some random factors. Is the tag making it easier to recapture them? Are the ones originally captured more stupid than most (and hence blunder into the nets again)? I remember one case where two blue wrens were recaptured, seven years later, in exactly the same place where they had been first captured and tagged, as yearlings. They simply hadn't changed territory or behaviour patterns. This sort of thing means that where mathematics meets zoology, life can become complicated and unmathematical.

So sampling with replacement, and sampling without replacement - it all begins with coins in the pocket.

And here's a cute brain teaser for small difficult children - you're in your bedroom and it's dark. You need to get a pair of socks from your sock drawer, but you can't turn on the light. You don't want to grab ALL the socks, you want to get as few as possible. In the drawer are loose grey and brown socks, all jumbled together. You don't care whether you get a pair of grey, or a pair of brown, just so long as you have a pair. All the socks are identical, other than these two colours.

What is the minimum number of socks you need to get, to be certain you have a pair?

Marg