Brain fuck: Chances.

[Nicknero]

One day me and my brother came in an argument about chances. We both had a different theory about how to calculate chances. And both theories may be considered valid.
This really blew my mind and I'd love to share it. See if I can blow your mind as well.

In the following example, we are going to play a little lottery game. And just assume that there are 10 million tickets for every lottery:

Theory 1:
You bought 1 ticket for the lottery. So simple common sense tells you that your chances to win the lottery is 1 out of 10.000.000, right?
But say you buy 2 tickets, for 2 lotteries (Since you can only have 1 ticket each lottery.) That means you have TWO chances to win the lottery, meaning your chances are doubled right? Because there are 2 lotteries, and you only have to win one of them.

Theory 2:
Continuing from Theory 1's story: Although you have 2 tickets for 2 lotteries, you forgot that because it are 2 lotteries, the amount of tickets in total are also doubled to 20.000.000 of which you have 2. Meaning that your chances to win at all is 2 out of 20.000.000, so that still equals 1 out of 10.000.000, meaning that your chances are still the same!

Argument from Theory 1 against Theory 2:
But Theory 2, you forgot the fact that there are TWO prices in the game now. So in that sense, you did double your chances to win!

Argument from Theory 2 against Theory 1:
No that's not true. You might think that there are 2 prices. But each ticket is only valid for 1 price. You can't win the first lottery with your second ticket. Neither can you win the second lottery with your first ticket.

See it as a dice game.
You have 1 dice. What are the odds of rolling 6 if you can roll only one time? Most people would say 1 out of 6, which is obvious. But what if you can roll TWO dice and only one of them has to be 6?
One would say that it doubles your chances seeing as you have two chances of rolling 6.
But another would say your chances are still the same, seeing as the total numbers you can roll are 12 now. But you still have two chances to roll 6. Hence, 2 out of 12 -> 1 out of 6.


Did I fuck with your mind yet? What do you think? Discuss! And let me find out what your way of chance calculation is.

[Statua]

Actually. There's a math behind this.

Dice for example. The probability of rolling a 6 is 1/6. If you're rolling 2 of them, the probabolity that both will be 6 is not 1/12, more of 1/36.

Probability is determined by Events/Number of outcomes. Probability of multiples is determined by multiplying all the events together.

Going back to dice, you have 2 events each with a 1/6 chance of hitting 6. 1/6 x 1/6 = 1/36

If you still dont believe me, take a look at how many ways you can combine numbers on 2 d6:

Spoiler for Hiden:

1-1
1-2
1-3
1-4
1-5
1-6
2-1
2-2
2-3
2-4
2-5
2-6
3-1
3-2
3-3
3-4
3-5
3-6
4-1
4-2
4-3
4-4
4-5
4-6
5-1
5-2
5-3
5-4
5-5
5-6
6-1
6-2
6-3
6-4
6-5
6-6

As you can see, there are 36 unique combinations you can make. Didn't think I'd actually use this shit I learned in high school.

So. Going back to your lottery. If you have 1 ticket for each lottery, it doesnt matter whether you got the tickets in the same lottory or a different lottery. You have 2 chances of 1/10,000,000 winning. So your chances are actually 1/5,000,000 at winning either or lottery.

NOW. If you're gonna get greedy and you want to win both lotteries, your chances of winning both A and B lottery are 1/100,000,000,000,000

Good luck

[Nicknero]

You missread Statua. When rolling two dices, they don't have to be BOTH 6. Just one of them.
So just taking your list of possible combinations, you see that 11 of them contain a 6. So you are saying that rolling 1 dice gives you 1/6 chance, and rolling two dices gives you 11/36 chance (1/3.2 chance)?

Now for the lottery part:
I said you can't have two tickets in one lottery. It's always 1 ticket per lottery. So indeed it are two times a 1/10,000,000 chance.
But how in the world does this mean winning either or lottery is a chance of 1/5,000,000?

You see how you these two examples are contradicting each other?
With the dice game, you say that rolling more dices LOWERS your chances.
With the lottery game, you say that playing more lotteries RAISES your chances.
While they both use the exact same theory. Explain?

[smt]

But say you buy 2 tickets, for 2 lotteries (Since you can only have 1 ticket each lottery.) That means you have TWO chances to win the lottery, meaning your chances are doubled right?

unless i am horribly mistaken, there is no increase in chance, you have the same chance of getting 1 winning ticket in one as you do the other, and it's equally likely you will get 1 winning ticket in both

[GamingZealot]

I have a different view on how chances work, based on how I was taught statistics. I'm going to start with the dice example because I am not entirely sure what you meant by two prices.

So with the dice example, what you are saying is that you start with a 1/6 chance of getting a 6 on a roll, and when you add another roll, it changes to a 2/12 chance, which reduces down to 1/6. The math being used behind this is that 1/6 + 1/6 = 2/12. This is not how fraction addition works; however, it is instead 1/6 + 1/6 = 2/6, reduces down to 1/3. That means your chances have indeed doubled.

The logic you are using where you are changing the possible results to 12 rather than 6 is the idea that you are increasing your possible roll results to 12, but in reality that is also wrong. This is where Statua's argument comes into play, that you are actually increasing the possible results to 36 not 12, because there are 6² as many possible results, not 6 * 2. This only matters when you are looking at possible combinations, though, and you stated that you are simply trying to get a 6 on one roll. This means that the chances that you will roll a six on either stays at 1/6.

Now the lottery example is the same case, although only because you stipulated that there are two separate lotteries. Assuming you only care about winning one and you draw from both, the chances are 1/10,000,000 + 1/10,000,000 = 2/10,000,000 which becomes 1/5,000,000. Had it not been stipulated that the lotteries were separate and you drew from the same one, it would instead be 1/10,000,000 + 1/9,999,999 = 2/99999990000000 reduces down to 1/499999995 (I think). This is because once you have eliminated one ticket from the pool via the first drawing, the second ticket has a slightly higher chance of winning (Assuming the lottery does multiple draws that is).

edit:
My explanation didn't seem quite right to me, because by the rules as I stated them, if you roll a 6 sided die 6 times, you have a 100% chance of rolling them, and rolling 7 times would give you over 100%. I haven't had any statistics training so the complex methods of adding probabilities isn't really something I'm good with, but a roundabout way of determining probability is to multiply the probability that it WON'T happen. For example, there is a 5/6 chance you won't roll a 6 each time you roll, so 5/6 * 5/6 = 25/36 which comes out to be just over 2/3, which when subtracted from one gives you the probability it will happen, which is just under 1/3, or 11/36.