Someone says “you have 1 out of 370 chances to win a bottle of water”
and you win 2 consecutive times in a row, what does that mean? i’m lucky or the probability is wrong?
In: Mathematics
You can’t tell if a probability is wrong from a single trial.
If something claims to have a 1/370 of happening, and you did it 370 times and you won every single time, then that probability is probably bullshit.
If you tried it twice, that’s not enough to determine anything.
There’s a lot of fairly complicated statistics you can do to work out how likely it is that you would see the results you’ve seen, assuming a certain probability. E.g, given that the chance of winning is genuinely 1/370, what are the chances that the first 2 results would be a win? And you’d probably discover that it’s unlikely, but not impossible, so you don’t have enough evidence to say that the estimate of the probability is wrong.
I would love to give a quick and simple explanation for how you do that, but there isn’t one, it’s complicated maths and any attempt to simplify it would be dishonest.
But, the important thing is: you can’t intuit it. Human brains are bad at interpreting probability. You can’t flip a coin a bunch of times and determine whether it’s fair by judging how fair it _feels_, what you do is write the results down and check if it actually is fair. I remember playing a dice-based video game, and I was confident that the dice were unfair and I was getting more low numbers than I should have. So, because I’m a nerd, I got out a pen and paper, and started noting the results… and it turned out I was wrong, the results were entirely fair, and I was noticing a pattern that didn’t exist.
It happens all the time, just try playing a board game with a superstitious person.
The winners are scrambled. So you can have 59 winners in a row, depending on how they are scrambled. For example, in almost all scratcher lotteries, you have a 1 in 3 chance of winning. Which means that if they print 300000 tickets, 100000 will be printed as winners. But there might be sections in the printing where they print 6 winners in a row. And there might be sections where there are no winners for 30 tickets. Comment if you need further explanation.
It means you are lucky. You’d have to play this game a great many times to by able to say anything useful about the probability being wrong, and even then you’d only be able to state it was wrong with a defined level of confidence.
Certainty is for children and pure mathematicians. For everyone else, there’s probability.
It’s not wrong because [we assume] the outcome of one event doesn’t affect any later outcomes. Being a 1/370 chance event multiple times in a row can occur, it’s just very unlikely. (1/370^2)
It probably just means you got very lucky.
If you spin a roulette wheel where the odds are 1 in 370, that means that people will win 1/370th of the time on average, “in the long run.” When we talk about long-run probabilities, it’s kinda like we’re imagining an infinite number of trials. If you spun it a million times, or a billion, or a trillion, then you would get closer and closer to 1/370th winning spins overall. The more trials, the closer a correspondence we expect between the odds and the actual outcomes. This is called the ‘law of large numbers.’
It might be easier to think about it in terms of coin tosses. If you tossed a coin 4 times, it’s entirely possible that you might get 3 tails and 1 heads. That’s a very ordinary outcome, not too wild a coincidence at all. Even though the theoretical odds are 50/50 and your results are 75/25. But if you tossed the coin 100 times in a row, it would be almost unthinkably unlikely for you to get 75 tails and 25 heads. As the number grows large, the likelihood of big deviations grows small.
There’s a *theoretical probability* – this is where you look at the possible outcomes of an event and decide what the odds of each outcome are. An easy example is a coin flip – it has two sides. On a flip of the coin, you have a 1 in 2 chance of getting one side or the other (50% chance). Or a die, with six sides. When you toss the die, you have a 1 in 6 chance of getting any one side.
Then there’s the actual or observed probability. This is based off of how many “events” or trials you actually run. So, if you flip a coin 10 times, you are actually likely *not* to get 5 Tails and 5 Heads, but maybe 6/4 or 7/3. Out of this real set of 10 trials, your observed probability of getting Tails might therefore be 60% or 70%.
*But*, the more trials you run, the closer to the *theoretical* probability you get. It’s called the *Law of Large Numbers*. So if you flip the coin 100 times, you might get Tails 55% of the time. Flip it 1000 times and you might get Tails 50.2% of the time.
Long story short, in your example, the person got lucky if they won twice in a row. Someone else playing this game could maybe go 800 times without winning once. But the longer the game goes on, the closer you’ll see the math work out to “1 in 370”.
edit: When they say “1 in 370 odds”, it could very well be that first theoretical probability. Say they printed out 5 winning tickets and 1,850 tickets total. When you simplify that you get 1 winning ticket for every 370 non-winning tickets. It would be possible for a person to get all 5 in a row but really, really, really, really, unlikely.
Probability is just pointing out the likelihood of something happening. Its not a set law that has to be obeyed. Its unlikely if you flip a coin 100 times you’ll get heads every time. But it can happen.
And interestingly using that coin analogy, the odds of landing heads 100 times in a row is 1,267,650,600,228,229,401,496,703,205,376 (according to google) That’s a disgustingly high number. But when you go to flip that coin the 100th time, it doesn’t matter what that huge number says , none of the flips before have any effect on this one right now. Its 50/50 no more no less.
Personally probabilities are a little weird. Like when they say you have a 1 in X chance of being struck by lightning, eaten by a shark, etc. They’re taking into account all of humanity versus the numbers affected. But if I live in in a land locked country and never go in the ocean how do i have the same probability of being eaten by a shark as someone who lives in south Africa and surfs every day? They can be misleading.
Its just luck. 1 in 370 twice is unlikely, but the second time you win isn’t effected by the first. Still 1 in 370.
This is an interesting question. If you knew for sure the chances are 1 out of 370 and you get to win 2, 10, 100 or wathever times in row you just got lucky enough because the *truth* is that the chances are 1 out of 370. But in practice how can you know this is the truth? At some point you have to ask yourself what is more likely, winning 100 times in a row or those odds being wrong?
And hence in statistics there’s aslo the concept of “the odds of the odds”. How likely it is that some odds are right given what you have observed so far. If you threw a coin 10 times and got 8 heads, you could say yeah it’s more likely for a biassed coin to produce this results but it’s quite likely for a fair coin as well, so there’s very little evidence that the coin is unfair. However if the coin comes heads 800 of 1000 times then the evidence becomes much stronger and there are ways to quantify this.
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