I was thinking about lottery odds, and how so much of the pitch is, essentially, you miss 100 percent of the shots you don’t take, with the thought that you should at least enter because your odds go up so much with just one ticket. The odds were non existent before, and now they exist even if they’re vanishingly small.
Is the difference between 1 in a million and 0 in a million actually somehow more than the difference between 1 in a million and 2 in a million, or between 492,368 in a million vs 492,369 in a million? Or are all three of these functionally the same?
In: 0
Let’s say there are three raffles you can buy, the 0 in a million raffle, the 1 in a million raffle, and the 2 in a million raffle.
You want to better your chances so you’re looking to buy enough raffles to get you a 10% chance of winning.
For the 2/million one you would need to buy 50.000 raffles.
For the 1/million one you would need to buy 100.000 raffles.
For 0/million, you cannot buy enough raffles or best case assuming its a rounded down 0/million you would need at least over 2.000.000 raffles.
So that means that they are not functionally the same.
You’re mixing absolute and relative odds. Buying two tickets doubles your odds and buying one “approaches infinitely” improves your odds (it’s division by zero which approaches infinity).
Buuuuut you still just have 2 tickets against a pool of millions of tickets.
If you want a head scratcher: imagine there’s 100 tickets total and you have 98 of them. There’s only 2 outcomes where you don’t win. If you buy 1 more ticket you’ve “doubled” your relative odds because there’s now only 1 outcome left (instead of 2). But absolutely you were already very likely to win because you had 98 tickets before. It works in a similar way when you have 1 ticket vs 2 except the absolute odds are you don’t win.
>What’s the difference between something with 1 in a million odds and 0 in a million odds?
0 in a million : impossible to win
1 in a million: you can win but rarely.
>Is the difference between 1 in a million and 0 in a million actually somehow more than the difference between 1 in a million and 2 in a million, or between 492,368 in a million vs 492,369 in a million? Or are all three of these functionally the same?
They are different.
The best way to look at these things is in terms of their proportions
Going from 1 in a million to 2 in a million is 100% better odds (2 times better)
Going from 492,368 in a million to 492,369 in a million is a 0.0002% better odds. (Basically no change in odds)
Going from 0 in a million to 1 in a million is an incalculable increase in odds (in some senses, more than infinity times better).
0.1 in a million to 1 in a million is 10 times the odds.
0.01 in a million to 1 in a million is 100 times the odds.
0.000001 in a million to 1 in a million is 1000000 times the odds.
Now as the 0.000…001 gets closer to 0, the odds increase when compared to 1 gets closer and closer to infinity.
However, 0 times infinity is usually undefined. So you can’t multiply 0 by infinity to get 1. (This is why I say in some senses it is better than an infinite times better odds)
You’re absolutely right. Buying the first ticket and buying the second ticket are identical bets (in a game where there are millions of tickets). Slogans like “you have to be in it to win it” are technically true in a trivial sense, but the first ticket is no more likely to make you a winner than the second ticket.
>Is the difference between 1 in a million and 0 in a million actually somehow more than the difference between 1 in a million and 2 in a million
In terms of calculating your expected winnings, no. With one ticket, you’ve spent, say, $1 for a one in a million chance of winning $500,000, so your expected winnings are 50c, and your expected return after the ticket cost is -50c. With two tickets, you’ve spent $2 for a two in a million chance of winning $500,000, so your expected winnings are $1 and your expected return is -$1. It’s the same bet every time, you’re just putting more money onto it.
Again, it is trivially true that if you don’t buy any tickets, you can’t win. But that doesn’t make the maths of buying one or ten or a hundred tickets any better.
The zero essentially makes the domain infinite. 0 in 1 is equally as unlikely as 0 in a million chance because there is no fundamental involvement. It’s almost like the antithesis of true certainty
The way I, a five year old, figure it, they are functionally the same.
If you buy zero lottery tickets, you have no chance of winning. If you buy 1 lottery ticket, you have a chance of winning, but it’s vanishginly small.
If you buy three of them, technically you have more of a chance to win the million, but practically your chances are still vanishingly small. Now if you were to buy one hundred lottery tickets, then you’d for sure win money.
But most likely not a million. Not even your money back. you spend 100 000 on lottery tickets, you’ll be lucky if you get 50 000 back. What a racket the lottery is.
Lotteries are a tax on people who don’t understand math.
Assuming you must pick 6 correct numbers ranging from 1 – 49, your odds of winning the jackpot are 1 in 13,983,816, per my local lottery website.
Sure, “somebody” has to win it, but you would be far better off just putting that money in an investment account that paid interest equivalent 5%.
In short, the difference between buying 1 ticket, or zero tickets, is that you will still have the ticket price in your wallet, a virtually infinite improvement in value vs. the ticket.
Imagine you have a nuclear plant. This nuclear plant has a flaw, every minute, it has 1 in a million chance to explode. How safe is it? The answer is not at all, because a “1 in a million every minute” means “it will explodes within 2 years in average”.
Imagine that that commercial planes had a 1 in a million chance to explode each travel. That would be roughly 30 plane exploding each year.
In other words, “1 in a million” matters when you deal with large scale. Large number of peoples, large amount of time, etc.
> Is the difference between 1 in a million and 0 in a million actually somehow more than the difference between 1 in a million and 2 in a million, or between 492,368 in a million vs 492,369 in a million? Or are all three of these functionally the same?
Numbers that are functionally the same in practice are numbers that have the similar “logarithm”.
For numbers bigger than 1, that means counting the number of digits in front of the decimal: so 1-9 is roughly the same, 10-99 is roughly the same, 100-999 is roughly the same, etc. Obviously, the limits are arbitrary, and if you want to be precise, 99 should be more similar to 100 than to 10, but that’s a good enough approximation.
For numbers between 0 and 1, that means counting the number of zeros before the first non-zero digit. So 0.1-0.9 is roughly the same, 0.01-0.09 is roughly the same, etc.
So 1 in a million and 2 in a million are functionally the same, but 1 in a billion would be fundamentally different.
The difference between zero and one is exactly infinitely greater than the difference between one and two.
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