Oddities

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Humans and not very good at calculating odds and how probabilities work. For instance, if we are tossing coins and we get six heads in a row, the probability of getting yet another head is still fifty-fifty. Yet people feel that after a series of heads that it is more likely that more tails than heads will turn up for a while, so that the ratio of heads to tails returns to the expected one to one ratio.

But the expected ratio of heads to tails for all subsequent tests is one to one. It’s as if a new set of tests is being started, and so any lead that has already built up is, in all probability, not going to be reduced.

This seems odd. If we have done one thousand trials and have turned up 550 heads to 450 tails, the ratio of heads to tails is about 0.818 and the ratio of heads to the number of tests is 0.55. Surely more tests will take the ratios closer to the expected values of 1.0 and 0.5? Surely that means that there will be more tails than heads in the future?

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Well, the answer to both questions is no, of course. The ratios for the whole test may move closer to 1.0 and 0.5, but equally, they may move further away. In the extreme case, there may never be a tail again. Or all the rest of the throws may result in tails.

Interestingly, if the subsequent tests produce a series of heads and tails, the difference between the number of heads and tails stays at around 100, but the ratio of tails to heads for the whole test slowly creeps closer to 1.0 and the ratio of heads to the total number of tests closes in on 0.5 as more and more trials are done. By the time we reach two million tests, the two numbers are not very far from the expected values, being 0.9999 and 0.5000 respectively.

So, if you think to yourself, as you buy a lotto ticket “Well I must eventually win, if I keep buying the tickets”, it doesn’t work like that. You could buy a lotto ticket forever, literally, and never ever win. Sorry.

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Lotto and sweepstakes are, I believe, a different type of gambling from other forms, such as betting on horses or poker and other gambling card games. Lotto, sweepstakes and raffles involve no element of skill, and the gambler’s only involvement is buying the ticket. Betting on horses or cards involves skill to some extent, and that skill comes down to things like working out the probabilities of a particular card coming up and the probabilities of other players having certain cards in their hands.

Both types of gambling encourage the gambler to gamble more. If a gambler doesn’t win on the Lotto he or she might say to his or herself “Better luck next time.” Of course, luck does not exist, but probabilities do, and this is a mild form of the Gambler’s Fallacy described above. Nevertheless, people do win and the winners appear on television for us all to see and emulate.

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There’s two sorts of strategy for winning the Lotto. First there’s the “always use the same numbers” strategy, and then there’s the “random numbers” strategy. If you always use the same numbers, goes the theory, then eventually there must be a match. That’s wrong of course, since the number combination may not appear before the end of the universe.

The random number strategy argues that there is no pattern to results so it is silly to expect a particular pattern to eventuate. This strategy acknowledges the random nature of the draw, but doesn’t give the gambler any advantage over any other strategy, even the same numbers strategy. It is certainly easier to buy a randomly generated ticket than to fill in a form to purchase the same numbers every time.

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Some people experience a run of luck. They might have three things happen to them, so go and buy a lotto ticket while their luck holds. Then is they win they attribute it to their lucky streak. It’s all nonsense of course. They conveniently forget the many, many times that they bought a ticket because of a lucky streak, only for the ticket to be a loser.

The proceeds from the sales of lotto tickets don’t normally all go to holders of winning tickets. Firstly the operators of the system need to recoup their costs. It’s not cheap to own and operate those fancy machines with the tumbling balls and it also costs to employ the people to check that the machines are fair.

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If one of the balls is dented, will that affect the probability of that ball being selected? Maybe, just a little, but the draw should be fair so those providing the lotto equipment spend a large amount of effort to ensure that they are fair, and the costs of that effort must come out of the prize funds.

Secondly, the state or maybe the lotto organisation itself will often withhold part of the lotto sales takings for local or national causes, such as cancer research, or societal things, like the fight against teen suicide. The money for humanitarian causes is deducted from the prize funds.

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One of the humanitarian causes is often the fight against problem gambling. It’s ironic and somewhat appropriate that funds from gambling are used to combat problem gambling. It seems that some people get such a thrill from gambling that they use all their, then borrow or steal from others to continue to gamble.

They invoke the Gambler’s Fallacy. They suggest that their luck must change sooner or later. It doesn’t have to, and may never change, but they continue to spend money on their gambling. They also don’t take account of the fact that they might win, eventually, by sheer chance, but it is unlikely that their winnings will cover what they have already gambled away. They have a tendency to believe that one big win will sort things out for them. It won’t of course.

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So, the only true fact about Lotto and similar draw is that you have to be in to win. But just because you are in doesn’t mean that you will win. You probably won’t. The best way to treat Lotto and other similar games is that you are donating to a good cause and you might, but probably won’t get something back. So, I’m off to buy a lotto ticket. I might win thirty million dollars, but I won’t cry if I don’t.

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What’s the probability?

 

transparent_die
Transparent die

We can do a lot with probability and statistics. If we consider the case of a tossed die, we know that it will result in a six about one time in six in the die is not biassed in any way. A die that turns up six one time in six, and the other numbers also one time in six, we call a “fair” die.

We know that at any particular throw the chance of a six coming up is one in six, but what if the last six throws have all been sixes? We might become suspicious that the die is not after all a fair one.

Dice
Dice

The probability of six sixes in a row is one in six to the power of six or one in 46656. That’s really not that improbable if the die is fair. The probability of the next throw of the die, if it is a fair one, is still one in six, and the stream of sixes does not mean that a non six is any more probable in the near future.

The “expected value” of the throw of a fair die is 3.5. This means that if you throw the die a large numbers of time, add up the shown values and divide by the number of throws, the average will be close to three and a half. The larger the number of throws the more likely the measured average will be to 3.5.

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Crap table

This leads to a paradoxical situation. Suppose that by chance the first 100 throws of a fair die average 3.3. That is, the die has shown more than the expected number of low numbers. Many gamblers erroneously think that the die is more likely to favour the higher numbers in the future, so that the average will get closer to 3.5 over a much larger number of throws. In other words, the future average will favour the higher numbers to offset the lower numbers in the past.

In fact, the “expected value” for the next 999,900 is still 3.5, and there is no favouring of the higher numbers at all. (In fact the “expected value” of the next single throw, and the next 100 throws is also 3.5).

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Pile of cash

If, as is likely, the average for the 999,900 throws is pretty close to 3.5, the average for the 1,000,000 throws is going to be almost indistinguishable from the average for 999,900. The 999,900 throws don’t compensate for the variation in the first 100 throws – they overwhelm them. A fair die, and the Universe, have no memory of the previous throws.

But hang on a minute. The Universe appears to be deterministic. I believe that it is deterministic, but I’ve argued that elsewhere. How does that square with all the stuff about chance and probability?

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Orbital

Given the shape of the die, its trajectory from the hand to the table, given all the extra little factors like any local draughts, variations in temperature, gravity, viscosity of the air and so on, it is theoretically possible, if we knew all the affecting factors, that, given enough computing power, we could presumably calculate what the die would show on each throw.

It’s much easier of course to toss the die and read the value from the top of the cube, but that doesn’t change anything. If we knew all the details we could theoretically calculate the die value without actually throwing it.

abacus
abacus

The difficulty is that we cannot know all the minute details of each throw. Maybe the throwers hand is slightly wetter than the time before because he/she has wagered more than he/she ought to on the fall of the die.

There are a myriad of small factors which go into a throw and only six possible outcomes. With a fair die and a fair throw, the small factors average out over a large number of throws. We can’t even be sure what factors affect the outcome – for instance, if the die is held with the six on top on each throw, is this likely to affect the result? Probably not.

Einstein's equation
E = mc2

So while we can argue that when the die is thrown that deterministic laws result in the number that comes up top on the die, we always rely on probability and statistics to inform us of the result of throwing the die multiple times.

In spite the seemingly random string of numbers from one to six that throwing the die produces, there appears to be no randomness in the cause of the string of results from throwing the die.

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Popcorn

The apparent randomness appears to be the result of variations in the starting conditions, such as how the die is held for throwing and how it hits the table and even the elastic properties of the die and the table.

Of course there may be some effects from the quantum level of the Universe. In the macro world the die shows only one number at a time. In the quantum world a quantum die may show 99% one, 0.8% two, 0.11% three… etc all adding up to 100%. We look at the die in the macro world and see a one, or a two, or a three… but the result is not predictable from the initial conditions.

Random
Random

Over a large number of trials, however, it is very likely that these quantum effects cancel out at the macro level. In maybe one in a very large number of trials the outcome is not the most likely outcome, and this or similar probabilities apply to all the numbers on the die. The effect is for the quantum effects to be averaged out. (Caveat: I’m not quantum expert, and the above argument may be invalid.)

In other cases, however, where the quantum effects do not cancel out, then the results will be unpredictable. One possibility is the case of weather prediction. Weather prediction is a notoriously difficult problem, weather forecasters are often castigated if they get it wrong.

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Lightning

So is weather prediction inherently impossible because of such quantum level unpredictability? It’s actually hard to gauge. Certainly weather prediction has improved over the years, so that if you are told by the weather man to pack a raincoat, then it is advisable to do so.

However, now and then, forecasters get it dramatically wrong. But I suspect that that is more to do with limited understanding of the weather systems than any quantum unpredictability.

Flooded
Flooded