Alternative calculation for average number of free games with retrigger
January 22nd, 2024 at 5:11:46 PM
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Hello Math nerds. Looking for some help here when examining free game awards and in particular, calculating the average number of free games when retrigger is possible.
Now I'm already familiar with the following way of calculating avg free games for a fixed free game award amount when you can retrigger FG awards within FG:
Let X = # of FG awarded when you trigger the FG win and let Y = the expected number of FG awarded per spin
Then Avg. FG = X + X*Y + X*Y^2 + X*Y^3 + ... = X / (1-Y). If Z = prob of triggering FG per spin then we can also view Y as Y = X*Z so that we get an overall formula of
X / (1-X*Z).
So if for example, you were awarded X = 10 FG per FG trigger and the probability of triggering FG per spin is Z = 1/50 then we'd have an avg FG total of
10 / (1-10/50) = 12.5 FG awarded when activating FG from the base game with potential for FG retrigger.
this makes sense intuitively, as you are initially awarded 10 FG and if you expect to trigger FG at a rate of 1/50 spins, or 10*(1/50) = .2 FG spins awarded per spin, then in those first 10 FG spins you'd expect to get an additional 10*10*(1/50) = 2 FG spins awarded and then from those 2 spins, you'd expect to get an additional 2*10*(1/50) more free games spins, and the process continues ad infinitum.
My issue is that I'm trying to conceptualize how you'd compute the avg free games when looking at it as the traditional expectation formula SUM( Total # of FG Awarded * Prob(getting that total # of FG) )
So using my example above it'll look something like Avg FG = 10*P(FG=10) + 20*P(FG=20) + 30*P(FG=30) + ...
I've been messing around with breaking them down into discrete cases depending on which spin # the FG trigger hits and then looking at them as a sequence of Bernoulli trials but despite my best efforts I either get a summation that diverges, or something that's close but still off from the way it's calculated above.
Does anyone see any reason why this secondary method of computation wouldn't work in theory?
Now I'm already familiar with the following way of calculating avg free games for a fixed free game award amount when you can retrigger FG awards within FG:
Let X = # of FG awarded when you trigger the FG win and let Y = the expected number of FG awarded per spin
Then Avg. FG = X + X*Y + X*Y^2 + X*Y^3 + ... = X / (1-Y). If Z = prob of triggering FG per spin then we can also view Y as Y = X*Z so that we get an overall formula of
X / (1-X*Z).
So if for example, you were awarded X = 10 FG per FG trigger and the probability of triggering FG per spin is Z = 1/50 then we'd have an avg FG total of
10 / (1-10/50) = 12.5 FG awarded when activating FG from the base game with potential for FG retrigger.
this makes sense intuitively, as you are initially awarded 10 FG and if you expect to trigger FG at a rate of 1/50 spins, or 10*(1/50) = .2 FG spins awarded per spin, then in those first 10 FG spins you'd expect to get an additional 10*10*(1/50) = 2 FG spins awarded and then from those 2 spins, you'd expect to get an additional 2*10*(1/50) more free games spins, and the process continues ad infinitum.
My issue is that I'm trying to conceptualize how you'd compute the avg free games when looking at it as the traditional expectation formula SUM( Total # of FG Awarded * Prob(getting that total # of FG) )
So using my example above it'll look something like Avg FG = 10*P(FG=10) + 20*P(FG=20) + 30*P(FG=30) + ...
I've been messing around with breaking them down into discrete cases depending on which spin # the FG trigger hits and then looking at them as a sequence of Bernoulli trials but despite my best efforts I either get a summation that diverges, or something that's close but still off from the way it's calculated above.
Does anyone see any reason why this secondary method of computation wouldn't work in theory?
