Dear mathematicians.

[tyggerjai]

So I'm playing this TLE game. And one of the things is sifting ore to find glyphs. And you can level up in sifting. Your base chance is 1% per 10K units of ore. That increases at 1% per level. Intuitively, it seems to me, that if I level to 10, I then have a 10% chance per run. If I make ten runs, I have a 100% chance of finding at least one glyph.

But someone has suggested that actually it's 81%, and waved something called Binomial Probability at me. So rather than looking it up, I'm asking the interwebs!

No, I did look it up briefly, but is this true? That actually there's some factorial thing going on here where my chances are reduced on each run?

And where do Rosencrantz and Guildenstern come into this? Aren't they dead!?

Comments

[(Anonymous)]

Your chances aren't reduced, but neither are they cumulative.

Try flipping coins. Each time you flip a coin you have a 50% chance of getting tails. This does not mean that 2 coin flips = 100% chance of getting tails. In fact, you have about a 75% chance. That's because each single coin toss is an unconnected event. It's possible to toss a coin 100 times and still not get any tails.

A grunty way to work it out is to look at all the possible outcomes. The possible outcomes for 2 tosses are HH, HT, TH, TT. 3 of those include T, so that's a 75% chance of tails.

If you roll a die, you have a 1 in 6 chance of rolling a 6. Again, you ain't necessarily gonna get a 6 just by rolling it 6 times. Again you can make a list of all the possible outcomes ...

111111
111112
111113

... etc, but it takes a bit more time to do that when you have more outcomes for each individual iteration!

If you were to be rather anal and sit down and write out all the possible outcomes for your ore sifting (given 9 chances of no and 1 chance of yes for each iteration) then yeh, it's gonna be less than 100%.

Of course, there are quicker ways of doing it. That's what binomial probability is all about. I'll leave you to look up the formula and crunch the numbers for yourself :)

[(Anonymous)]

Oh - and that was me, Halo. CBF logging in via OpenID :-P

[tyggerjai]

81%, apparently.
I'm sure I used to know this stuff. *sigh*

And I knew it was you :)

[(Anonymous)]

How did you know it was me? K8's equally maths nerdy!

...

And has a DW account. Dammit.

[tyggerjai]

And Thorf and Lederhosen also. It wasn't the maths that gave you away. It was the "Ain't necessarily gonna".

[(Anonymous)]

Also, 81% seems a bit high to me ...

[(Anonymous)]

If you have 10% chance of finding a glyph per run, then you have a 90% chance of NOT finding a glyph.

If you then go on N runs, the chance of you finding zero glyphs is 0.9 raised to the power of N.

(0.9)^10 is roughly 0.35.

This means that after ten runs at level 10, you have roughly a 1/3 chance of having found nothing, and a 2/3 chance of having found one or more glyphs (the exact probabilities for each number of found glyphs can be worked out if you really want).

Level 10
RUNS % chance of finding one or more glyphs
0 0
1 10
2 19
3 27
4 35
5 41
6 47
7 52
8 57
9 61
10 65

For any level L, the probability of finding zero glyphs after N runs is ((100-L)/100)^N. Basically, it's never going to be exactly 100% unless you're already at Level 100 to start with.

[thorfinn]

10% per run means that in X runs, as X approaches infinity, you will retrieve X/10 glyphs. The more actual runs totalled, the more likely you are to reach that that average. But, for more fun yet, the history doesn't matter. If you are below or above average on runs currently, you will *not* have more chance of approaching the average than someone else who has no runs. That's how Rosencrantz and Guildenstern come into it. :-)

The probability of "finding at least one glyph" is better expressed as 1.0 - "the probability of finding no glyphs".

See Anonymous Two for the details, they are right. :-)

Anonymous One a.k.a. Halo has it right too. Repeat event probability is multiplicative, not additive, but you have to multiply the right thing.

[(Anonymous)]

Because everybody else has already given the right answer but I still need to stick an oar in...

One trick I find useful for understanding this result is to think about the probability of multiple events:

If I have a 10% chance per run, and I do 10 runs, my average payoff is 10 x 0.1 = 1.

But it's clearly possible to get 2+ finds in 10 runs - so there must be some 0s to make the average work out to 1.