If you have 2 people working on something that has a 1 in 10 million chance, does the probability increase as a whole

Question

I was talking with a friend who was grinding for an item in a game. The item has a 1 in 10 million chance of dropping. He said that he should get his friend to help him mine for it so that he has a higher chance. We then had an argument where I stated that you will still have a 1 in 10 million chance of getting it, even if 2 people are working on getting that item. I said this because I based it on the fact that if you have a 25% chance of getting an item, you can try 3 times and still end up with a 25% chance. Even though it might look like you now have a 75% chance. Its probably a dumb question but I'm genuinely stumped on this.

Answer 1

I presume that the "1 in 10 million chance" is the chance per instance of opening a package in the game.

I guess you would agree that opening only one package, your chance of getting the item is less than if you open two packages. Actually if you open $n$ packages, your overall chance of success will be $1-\left(1-10^{-7}\right)^n$. Note that for $n=1$ this is $0.0000001$, and for $n=2$ it has not quite doubled: $0.0000001999999899\ldots$.

If you open a million packages, with $n=10^6$, you get something like $0.095$ chance. So you can see that with more openings, the overall chances of at least one success are rising.

If you help your friend, you are roughly doubling the rate at which packages are opened. So you are letting that $n$ grow twice as fast. If you each do a million openings, you raise that previous chance to about $0.18$ in the same amount of time as it would have taken your friend to raise the chances to $0.095$.

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I think I would add that in one sense you are both correct. If you have already opened one million packages, and thus far you have failed to succeed, then opening that next package all on its own still has a one in ten million chance of containing your item.

But at the moment in time when your friend asks you to join in, you don't yet know that the first million times will all fail. So it's not right to think that way yet.