Life After Full-time Work Blog

Here’s where we can consider Joe DiMaggio, Manchester United and Bill Gates

Last time we looked at all the possibilities when we toss a heads-or-tails coin. Maybe it’s an unbiased coin (as likely to fall H as T), or maybe something has been done to the coin to make (let’s say) H more likely than T. It’s really tedious, but we can (in principle) list every possible combination of outcomes in, let’s say, a series of 10 tosses, as well as the relative chance (or “probability,” to use the jargon term) of any specified combination of Hs and Ts. We don’t have to actually list every combination, or know the formula that gives us the probability of each possible combination. Let’s leave that to the professional mathematicians. For us, all we need to understand is that typically many outcomes are possible, some of which are extremely unlikely.

Do we find stuff like this in life? We don’t actually know! In our examples last time we knew exactly what we were doing (tossing a coin many times), and we knew there are many possible outcomes. What happens in real life is that we see some sort of outcome, and we have no idea whether that’s the result of chance or something else.

If we’re deeply interested, what do we do? Well, if we’re mathematically inclined (as I am!), we might build a simple model of some kind, that tries to capture the essence of the outcome we’re interested in, and see, according to that model, what is the chance of that outcome. And then we might believe, or not believe, that luck caused it – but we can never know for sure because we just observed one outcome.

For example: Joe DiMaggio.

Back up a moment. Who was he (apart from being Marilyn Monroe’s second husband)? He was an extremely talented and famous baseball player for the New York Yankees, and in 1941 he set an incredible record of getting a “hit” in 56 consecutive games – never done, or even approached, before or since. (The second longest sequence in the modern game is Pete Rose’s 44 hits.)

And so the obvious question: could DiMaggio’s streak have just been luck?

Here’s the model I built. I know that his batting average over that sequence was roughly (an amazing) 0.400 (that is, a 40% chance of getting a hit, every time he came to the plate). How many times in a game would he be involved in this way? On average, roughly 4 times a game (the other times, if he came to the plate more than 4 times, he might get a walk or be hit by a pitch or have some outcome that didn’t affect his 0.400 average).

OK, assume (in the model that I’m building) that every plate appearance gives him exactly the same 40% chance of a hit (regardless of what happened in his previous at-bat). That means there’s a 60% chance that he won’t get a hit. So the chance that he has 4 plate appearances, all without a hit, is (0.6)x(0.6)x(0.6)x(0.6), which amounts to 0.1296: roughly 13%. So, in the remaining 87% of cases, he gets at least one hit. And that’s the starting point we need.

What’s the chance that he gets a hit in 56 consecutive games? Assuming (as I do) that there’s the same 40% chance of a hit at each plate appearance, he’ll have that 87% chance of at least one hit per game, so the chance of 56 consecutive games with at least one hit in each is 0.87 multiplied 56 times – or (keeping lots of decimal points!) 0.000421.

But wait: in a season of 154 games, that 56 game hit streak could, in chance, start with any of the first 99 games. So the chance of that streak occurring at some point in the season is 99 x 0.000421, which is 0.0417: a bit more than a 4% chance. Gosh, that’s huge! It implies that, for any one player who can average 0.400 over a long period, we’d expect to see such a streak occur on average once every 24 years. Yet it has never ever happened, apart from Joe DiMaggio in 1941: no other player, no other year.

That suggests that hitting 0.400 over a long period of games is highly unusual. (As a Toronto Blue Jays fan, I remember John Olerud in 1993, carrying a 0.400 batting average into the start of August, around 120 games. And that was the first time since Ted Williams in 1941.)

Would a .300 batting average do it? In a word, no. It gives you a 30% chance of a hit in any one at-bat, a 76% chance in any one game, and an expected 56-game streak once every 48,000 years. Pete Rose hit 0.385 over his 44-game streak in 1978 in a 162-game season. Anyone hitting with that consistent average should get a 44-game streak every 7.5 years: again, to put it mildly, that’s obviously an average rather difficult to sustain!

My conclusion? These streaks are highly unlikely to appear simply as the result of luck. Yes, luck is obviously involved (as it must be, with DiMaggio’s 13% chance of not getting a hit in any one game), but there’s surely a huge amount of skill involved. The luck is that it happened to those players, when there were so many others who approached the same sort of averages over the same sort of periods (like Willliams and Olerud, for example), but had at least one no-hit game in their own excellent streaks.

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On to Manchester United, whom I support. (It’s a long story why, since I’ve never lived in Manchester.) Soccer (or football, as it’s known in many countries) is an interesting game in which scores are typically very low. In baseball, scores are higher; in basketball, much, much higher. What this implies is that, if one team is superior to its opponent, they’re much more likely to win against their opponent in basketball than in soccer. In basketball a good team has so many chances to score that, even if they missed many, they’d score a large number of points, so any gap in skills would have a much bigger chance of showing up in the final result than in soccer, where two goals in a game is pretty good going.

(I look at the standard deviation of winning percentages in top leagues in basketball, baseball and soccer, and you find it’s much bigger in basketball than in baseball, which exceeds soccer. Not a surprise: the scoring systems are much more likely to reward skill differentials.)

In soccer they have a statistic that they call “expected goals”: when a shot is taken from that particular field position, with that particular number of players between the shot and the goal, what is the probability of a goal resulting? For example, if one shot in ten gets a goal in those situations, the expected goals resulting is 0.1 (called “xG = 0.1”). At the end of a game, you might get one team with 1.2 xG and the other team with 0.9 xG, implying that if the game were played the same way 100 times, the first team would expect to score 120 goals and the second team 90 goals. But of course in a single game the first team might win 3-0 or lose 0-2: totally consistent with the xG’s. There’s obviously a lot more luck involved in the outcome of competitive soccer games than in other sports, because soccer scores are so low. A 5-0 result in soccer is an outrageous win; a team that only scores 5 points in a basketball game is unheard of.

(Side observation: I frequently watch live TV coverage, and I’m always amused when a commentator shouts something like: “An open goal – he was never going to miss that!” Yeah, right! Remarkably often a ball is skied above an empty net, or something else like that happens. It shouldn’t be a surprise, though, and it’s hardly a reflection of skill, or lack of it. These are all hugely skilled and talented players, but it’s pure chance if a quick pass to them is a few inches higher, or perhaps lower, than would be ideal, or if their quick kick is angled 5 degrees higher, or 5 degrees lower, than would be ideal. It’s pure immediate reaction, no time to adjust to ideal conditions, or they’d be robbed of the ball. That’s why xG’s are so often so low.)

Anyway, back to Manchester United. In the 2023/24 Premier League season, they finished in 8^th place out of 20 teams, with 60 points. Aha! But let’s look at their xG statistics. They actually scored 57 goals, as compared with their xG of 60; but more importantly, they only conceded 58 goals, as compared with their xG of 75. This wasn’t the result of exceptionally good goalkeeping, because both their goalkeepers came in for much criticism; it was just exceptional good luck.

What I’m saying is that, based on the xG model, they should have finished 16^th (not 8^th) out of 20 teams, and should have earned 40 points rather than 60.  I’ll add that I wasn’t entirely surprised when, in the following season, they actually finished 16^th – and they fired their manager.

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Where does Bill Gates fit into all of this? He doesn’t – at least, not directly. But he’s one of the exceptionally wealthy people in the world (and he shares his wealth generously). We’ve all seen the statistics about wealth, about how half the world’s wealth is owned by the top 1% of the population, the top 10% own 76% of the aggregate wealth, and the bottom 50% of the population together own 2% of the wealth. Yes, an extreme distribution. Hence my question, in the context of these blog posts: can this sort of extreme distribution simply be the result of luck?

There’s actually a paper I found on the subject, and its answer is: not totally explainable by luck, but yes, to a surprisingly large extent. And yes, the authors reach that conclusion by building a model.

They start by assuming a distribution of talent in the world. They use what’s called a “normal” distribution (that’s the technical name: there’s nothing normal about it!), with most people close together in talent, but some very talented and some very untalented. They consider a working life of 40 years, with everyone starting with the same amount of capital. Then, every 6 months (so, 80 times over a working lifetime) events happen, with equal chances of good and bad luck. If someone has bad luck, their capital is halved. If they have good luck, their capital may or may not double, the probability being proportional to their talent. They use 500 events for a population of 1,000 people, so at any event, half the people are left untouched, a quarter experience good luck and a quarter bad luck. That’s the (obviously artificial) model they use.

After 80 such cycles, what is the distribution of ultimate wealth? That’s a single iteration of the model.

It turns out that you end up with a very large number of poor people and a small number of very rich ones.

Among the interesting side results, I saw that about 16% of the people had neither lucky nor unlucky events over the 40-year period, and 40% experienced events all of the same type, that is, either all lucky or all unlucky (talk about a good streak or a bad streak – over a lifetime!). The most talented people turned out not to be the most successful ones. In fact, the most talented person ended up with slightly below average wealth; the one with the most wealth was only moderately talented. 20% of the population ended up with 80% of the wealth. (In real life, you’ll remember, 10% own 76% of the wealth: not identical to what the model predicted, but either way, a huge concentration of wealth.)

Yes, as I said, this is a very artificial model. It’s not in any way how the world works. And I’m guessing that the authors performed many more simulations with parameters different from those described a few paragraphs earlier, and settled, for publication, on the set of parameters that best created an outcome close to the actual worldly state of affairs. But it does show that luck superimposed on talent (remember Joe DiMaggio?) can mathematically explain a huge variation in outcomes over even just one lifetime.

They add that they ran the experiment with many different starting conditions. No big surprises in the outcomes. Their very generalized summary: “ … in spite of its simplicity, the … model seems able to account for many of the features characterizing … the largely unequal distribution of richness and success in our society, in evident contrast with the [normal] distribution of talent among human beings. At the same time, the model shows … that a great talent is not sufficient to guarantee a successful career and that, instead, less talented people are very often able to reach the top of success …”

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Takeaway

These are examples that demonstrate (at least to me) that luck plays a huge role in our lives.

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And so, next time: what does all of this suggest for the way in which we might consider living our lives?

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