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Gambling has been mentioned already, but I think it's also the statistics of gambling that gets lost on people. If something has a 1 in 30 chance of a payout, it doesn't mean that in 30 tries there will be at least one payout, it means that there is a thirty percent (I don't know the percentage accuracy right now) chance of that single attempt to payout. When I worked in a liquor store and sold scratch off tickets, people would look at the odds on the back and buy so many thinking this way.
Yup. There's no number of scratchers you can buy that gives you a 100% chance of winning. Sure, your chances go up the more you buy, but it never reaches 100%.
The formula is:
1 - (1-p)^Nwherepis the chance of winning andNis the number of scratchers you buy. Basically, you have to NOT win for N scratchers, so we multiply (since this is an AND condition, ie: you must lose scratcher A and scratcher B and scratcher C, etc) the chance of not winning (1-p) by itself for the number of scratchers bought. That's the overall chance of not winning, so we subtract that from 1 to get the chance of winning. You could instead use the chance of winning directly, but the formula is much longer (until you simplify the equation, which would give you the same answer as above) since you'd need to add (in this case we are using OR conditions) the chances of winning 1 scratcher or 2 scratchers or 3 scratchers, etc.1 in 30 is a 3.33% chance of winning (a 96.67% chance of not winning, for those still following along). If you buy 30 scratchers, your chance of winning is only 63.83%. For 300, it's 99.9962%. The chance will never reach 100% because you have a number between 0 and 1 raised to the power of a positive number in the formula. The chance of winning at least 1 of N scratchers can only be 100% if the chance of winning a single scratcher is already 100%, and they don't sell those.
However! There are rules dictating the distribution of winning scratchers in a roll. It's obviously not 1 every 30 exactly, but it's also not perfectly random (which could lead to long strings of losing scratchers or long strings of winning scratchers). That's why sometimes you'll have to wait in line behind someone while they make the gas station attendant open a whole new roll because they want to buy 100 contiguous scratchers and there were only 99 left in the old roll.
Turns out, humans don't think true randomness "feels" random. There's actually a game design trick where you tell the player odds that are lower than reality because the true odds "feel" lower than the reported number. Pokemon did not use this trick, so Hyper Beam (reported accuracy of 90%) feels unfair, since you remember more strongly all the times it missed when you really, really needed it to hit vs. all the times it hit.
If you buy all the scratchers you will win, you probably wouldn't net a positive amount, but you would win.
Ah, yes, bask in the "technically correct"ness. 🤌🏼