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Suppose 500,000 lotteries were played before yesterday. Yesterday, Jane played the lottery and won. Today, she played the lottery and won again.

Person A, let's call him the conspiracy theorist, might say: "This is incredible. How did she win two lotteries back to back? The probability of that seems too low. This must have been rigged.

Person B, let's call him the skeptic, says: "There is no need for surprise. 500,000 lotteries have been played. Sooner or later someone had to win two lotteries, even if it was back to back."

Who is correct here? Many would say person B is correct but why? Each lottery is independent. The fact that 500,000 lotteries were played before has no influence on the probability of the lottery wins today and yesterday.

Let's now shift the scenario a bit. Let's assume no lotteries have ever been played before yesterday. The first day the first ever lottery is played yesterday, Jane wins. The second day it's played, today, Jane wins again. It seems, intuitively, that the chances of the lottery being rigged is now much higher here. And yet, the probability of Jane winning today and yesterday is clearly the same regardless of how many lotteries were played before.

One might say that the reason why this intuition occurs is because a person/agent/etc is more likely to rig the first two lotteries out of two ever played than at the 501st and 502nd lottery and not anywhere else. But how can we make this assumption? Let's assume for argument's sake though that a rigger is just as likely to rig the first two lotteries than say the 501st and 502nd. In this case, are our intuitions wrong?

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  • It seems you are trying to determine whether a lottery is rigged based on only two consecutive outcomes. Probability analysis work best with a very large data set. Two outcomes is not enough information.
    – user59124
    Nov 29, 2022 at 15:40
  • There is not enough context to infer if this event is unlikely. For example, how many tickets were sold in each lottery? How many did Jane buy? And so on.
    – Boba Fit
    Nov 29, 2022 at 16:32

3 Answers 3

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First of all, that depends on the probability of winning the lottery. Many lotteries have probabilities that are so ridiculously unlikely that 500,000 plays means nothing. Seriously, it's not uncommon to have odds of 1:100,000,000 and therefore 500,000 isn't even enough to, on average, expect 1 winner, let alone 2.

However if your lottery has a win rate of 1:100 then winning twice would be akin to rolling a dice with 100*100 =10,000 face (1 for each combination). And for 500,000 tries or 250,000 two roll tries you'd expect to see that on average 25 times.

Now "on average" is a weasely concept, because these are independent events and it's just the average of large numbers of trials that are predictable, but a single event or even a low number of trials could happen more or less often than that. The expectation value just says that with N tries and a probability of 1/N you can expect to see the result once "on average". However it doesn't say whether it's on the first or last try or twice in the first N rounds and none in the second and so on.

So for normal lotteries your "conspiracy theorist" would have a strong point that odds of 10^16 or 1:10,000,000,000,000,000 are so unlikely that a rigged process is more likely than a fair game. Though in the end unlikely doesn't mean impossible, which is why statistics has to manage two kinds of errors one where you accept the hypothesis (that the game is fair) despite it being unfair and the other where you reject the hypothesis despite it being fair.

And where you set these limits is essentially arbitrary and more of an expression of what you'd like to avoid more as shrinking the margin of error on one increases it for the other. So for example for tests for diseases you'd rather have false positives than false negatives. So better more people who are marked as diseased and can be further tested than having people marked as healthy that are actually sick and could spread the disease without knowing. While if you produce idk high quality products you might rather reject good products than risk having a faulty one on the market. But again with 10^16 against you even conservative error margins would probably reject that.

In terms of what sounds more likely to be rigged, well I guess it's more of a psychological thing. Like while correlation doesn't equal causation and while short term trends don't have to be long term trends, we still have a tendency for pattern matching and if things happen at the same time or if we see a trend we are likely to assume a connection or a larger pattern (which is what bites gamblers quite hard).

Like in the case of winning the first 2 lotteries it's a winning streak of 100% that we suppose will continue. While in the second it's winning 2 out of 502 lotteries which is a better than expected win ratio but still a pretty bad one. Like if you started playing now it would, on average, take you a year to win twice even with these odds being massively more in your favor than the actual ones.

Though if you want to rig a lottery - and let's be real all lotteries ARE rigged against you - you wouldn't do it by winning multiple times you'd just set the jackpot slightly lower than ticket price * expectation value. Because that way you actually win the lottery with every game because on average you net more than you give out. And that's something that you can plan with some reliability (for large numbers), while the individual game is pure chance.

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Yes. We may THINK an event is "chance driven" but we could be wrong. It may be pseudo-random, with some glitch, or it may be hackable, or pre-wired by the holders of the event.

The operators of a lottery try to hack-proof, and test for pseudo-randomness, but for sure if their first two winners were the same person, they would do a deep dive into all their processes, plus surreptitiously investigate the winner. Likewise, if there is a free press, it would do an investigation into the link between the winner and the organization holding the lottery, or any of its managers.

Decades ago, when I was in college, my philosophy prof in one seminar class asked our class whether we would bet money at 1:1 odds against someone rolling a 6 on a die. He then asked "what if he already rolled a 6". And what if he has already rolled a 6 10 times in a row."

I was willing to take that bet in the first two instances, but not the third. One of my classmates would still have taken the bet with 10, or a hundred, 6s in a row. He was confident that the process WAS random. This is the view of a theoretician. As an empiricist, I went with the observations instead. The process was not random. Investigating HOW it as not random was not something we could do in the proposed scenario, but we didn't need to know how the cheating was happening to save our money.

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People don't think of it any of the times it isn't back to back winning the lottery. Say there was a 1 in 300,000 chance of it being back to back. If it happened the first two times, then it would be noticed. But they aren't noticing it the other times. If they picked those two times in particular, it would be noticed more if it happened because it had a chance of being noticed failing. But people normally won't notice the times it doesn't happen but will notice the times it does happen.

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