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I have 1000 rolls and I can get 4 rewards:
Reward A - 30%
Reward B - 14.4%
Reward C - 40.6%
Ability to re-roll - 15% *re-roll has the same chances for rewards, so it can give a re-roll again.

I want to calculate final % chances for rewards A, B and C, so they sum up to 100%.

My only idea was to put this to some kind of algorithm and loop a lot of times, so the result is really close to 100%.

Are there any maths formulas for a given problem? T

I tried to multiply all chances by 1.15, but this doesn't give the correct final result.

JMP
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XPOLKYT STUDIOS
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1 Answers1

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Event $A$ occurs when we throw one of $A, XA, XXA, \dots$, where $X$ is the re-roll.
Event $B$ occurs when we throw one of $B, XB, XXB, \dots$.
Event $C$ occurs when we throw one of $C, XC, XXC, \dots$.

So each event occurs as:

$$A(1+X+X^2+\dots)$$

The term in the brackets is the well-known geometric series, and has the closed-form sum of:

$$\frac{1}{1-X}$$

$X$ is this example has the value $\frac{15}{100}$, and so the geometric series is equal to $\frac{1}{1-{\frac{15}{100}}}=\frac{100}{85}$.

Multiplying each probability by this amount gives you the actual likelihood of each event (with the likelihood of the event of an infinite number of re-rolls tending to zero).

i.e.

$A \approx 35.294\%\\ B \approx 16.941\%\\ C \approx 47.765\%$

JMP
  • 21,771