Life After Full-time Work Blog

It’s tedious, but possible, in some cases to list all the possibilities – and that teaches us many lessons

 

Call it luck, call it coincidence, call it fate, call it the unseen hand of the person above – whatever you call it, most of us have at some time experienced something totally unpredictable and out of our control, that changes some aspect of our life, and takes us on a different path.

In fact, I think stuff like that happens all the time, but we don’t think of it that way, and we don’t notice it, because usually it doesn’t have much of an impact. But our lives have so many dimensions – family, friends, work, community – and so many people are involved who do things their own way, that we can’t possibly expect everything to be predictable and controllable. It’s only the big changes that we notice.

I’m going to call it luck, though you can of course think of it differently, no problem. And I mention all this because occasionally I start thinking about this sort of thing, and wondering to what extent these things determine how our lives evolve – and then, not only the consequences to us, but how the distribution of things in the world evolves. In other words, not just how an individual’s life develops, but also how aspects of the world develop.

And, given my mathematical background, I’ve wondered if it’s possible to make some estimate of the variability of outcomes that luck can cause. So I occasionally notice something that I’ve read that involves those thoughts, and since I recently came across two such things and was reminded of a third, I’ll tell you what this has to do with Joe DiMaggio, Manchester United and Bill Gates. And then about the lessons this has taught me about how to live some aspects of life.

But first, let me start at the start, which involves how luck operates. Some of you will immediately think: “Oh no, there’s going to be mathematics involved, and I’m not only hopeless at math, I’m afraid of it.” Fear not! You use numbers every day in your life, without ever needing to understand the math underlying them, in the same way that you use words every day without having to analyze the grammatical structure of your sentences. You’re comfortable with speaking; you’ll get the math of luck very easily. There isn’t any formula involved, to scare you!

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Here’s an example about how luck operates, and its range of possible outcomes and how likely or unlikely those outcomes are. It’s a deliberately simple and artificial example, which I’m using just to illustrate some useful principles. It has to do with tossing a coin: a head or a tail, with exactly the same (50%) possibility of each of those outcomes. Bear with me, I have a reason for using it.

Suppose we toss the coin once. There are two possible outcomes, which I’ll call H and T. Let’s posit explicitly that this is an unbiased coin, not artificially weighted in some way to be fractionally more likely to favor a head (or a tail, for that matter). So the chance of H is 50%, and the chance of T is also 50%.

If we toss it a second time, we get the same two possible outcomes. So the sequence of two tosses could be HH, HT, TH and TT, each equally likely (25%). So there’s a 25% chance of two heads, also a 25% chance of two tails, and a 50% chance of one head and one tail (if we don’t care about which one comes first; we’re just counting the numbers).

Let’s suppose we toss the coin 10 times. Thankfully, I don’t have to manually extend those lists (like HTTHTHHTTH and so on): there’s a formula that gets us to the overall possible outcomes (and you don’t need to know what the formula is – trust me). It turns out that, out of 1,024 possible sequences of outcomes of heads and tails:

1 contains exactly 10H and 1 contains exactly 10T (roughly 1 chance in 1,000 of each);

10 outcomes contain 9H and 1T; 10 contain 1H and 9T (roughly 1 chance in 100 of each);

45 contain 8H and 2T; 45 contain 2H and 8T (just under a 5% chance of each);

120 contain 7H and 3T; 120 contain 3H and 7T (roughly a 12% chance of each);

210 contain 6H and 4T; 210 contain 4H and 6T (just over a 20% chance of each);

252 contain 5H and 5T (roughly a 25% chance).

A couple of observations.

The likeliest (by which I simply mean the most frequent) outcome is 5H and 5T, not surprisingly. But maybe “likeliest” is a misleading word to use, because in another sense it’s not actually likely, because of the 1,024 sets, although 252 have 5H and 5T, the other 772 possible outcomes have some other combination of Hs and Ts. So if we do a series of 10 tosses, it’s more likely that there won’t be exactly 5 each of H and T, even though 5 of each is what we might think of as the “expected” (in the sense of “most natural”) outcome.

An extreme outcome (all 10 the same, or 9 out of 10 the same, or whatever we think of as extreme) is highly unlikely – even though of course it’s still possible.

And so we can conclude that, if we were to perform thousands of combinations of 10 coin tosses, they’d converge overall toward roughly equal numbers of Hs and Ts, but that we’d also expect to occasionally get some bizarre elongated sequences. Yet that possibility (though it exists) is so small that we’d disbelieve it, in practice. In fact, if we only performed one sequence of 10 tosses, and if we got an extreme number of Hs or Ts, we’d be much more likely to think: “Hey, surely that’s a biased coin.”

Congratulations on surviving your first lesson in the mathematics of probability theory!

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I want to take this just one step further.

Suppose we really did have a biased coin, one that was finely chiseled in a way that a toss could still come down either H or T, but it’s 60% likely to come down H and only 40% likely to come down T. (In our example, you’ll remember, those likelihoods were both deliberately set at 50%.)

How would the 60/40 substitution change the chances of those outcomes after 10 tosses?

Again, trust me on giving you the numbers. You’d expect the chance of Hs to be a bit bigger than the chance of Ts, intuitively. And your intuition would be right. We’d find:

10H and no T: a bit less than 1% of the time;

9H and 1T: roughly 4% of the time;

8H and 2T: roughly 12%;

7H and 3T: a bit more than 21%;

6H and 4T: roughly 25%;

5H and 5T: roughly 20%;

4H and 6T: roughly 11%;

3H and 7T: a bit more than 4%;

2H and 8T: roughly 1%;

1H and 9T, or 0H and 10T: an absolutely tiny chance.

I’ve shown all those numbers simply for completeness; they have no independent significance. But let me make a couple of observations, again.

The most frequent outcome is 6H and 4T, not a surprise because our 60/40 probabilities make this the natural “expected” outcome. But, as before, “a combination other than 6H and 4T” is more likely than 6H and 4T.

And an extreme outcome (9 or 10 H or T) is extremely unlikely – though now, with H occurring 60% of the time, getting “9H and 1T” actually has a 4% probability – meaning that if you waited for 25 sequences of 10 tosses, it would be reasonable to find that one of them contained 9H. That extreme event is slightly more possible now, because we increased the chance of getting H from 50% earlier to 60% in this example.

That’s it. That’s your second probability lesson completed. Congratulations!

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What if we tossed dice instead of a coin? Then we’d have 6 possibilities each throw, instead of 2. Gosh, that would make the tabulation of outcomes extremely tedious! But it could be done, of course (by someone else, not by me!).

What if we drew cards from a pack? Oh gosh, even more tedious! But again, it could be done.

What if we had probabilities that weren’t the same for each draw? In principle, not a problem. We could keep making things increasingly complicated, but it’s obvious we’d always come to some conclusions. One is that there’d be a huge range of possible sequences. Another is that extreme sequences are always possible, but unlikely, so that, if we saw one, we’d have difficulty believing it.

And that’s it. That’s all the probability theory you need.

How does this apply to real life? That’s for the next post.

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Takeaway

With any set of chance events (not necessarily coins or dice or cards), it’s possible (if we actually know the initial probabilities) to calculate the chance of every possible outcome. Most of the time the outcomes are reasonably close to what we’d intuitively “expect”; very, very occasionally there could be an extreme outcome – and then, if we actually came across it, we’d naturally be very suspicious of it!

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