Expect number of free spins
October 23rd, 2015 at 6:42:07 AM
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How to calculate the expect number of free spins if free spins can trigger free spins infinitely?
In https://wizardofodds.com/games/slots/atkins-diet/ in bonus section it just states a formula that is not clear to me, where it came from.
In https://wizardofodds.com/games/slots/atkins-diet/ in bonus section it just states a formula that is not clear to me, where it came from.
October 23rd, 2015 at 4:56:39 PM
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Here's how it is done in the Wizard's example:
Let S be the expected number of free spins generated from a particular spin. In the Wizard's example, the probability of generating free spins is 0.011185, and you always get 10 spins, so S = 10 x 0.011185 = 0.11185. Note that if S >= 1, then the expected number of free spins is infinite.
The expected number of free spins from the S spins = S x S.
The expected number of free spins from the S x S spins = (S x S) x S.
And so on.
The total number = S + S2 + S3 + S4 + ...
Since S < 1, this equals S / (1 - S).
(Proof: multiply (S + S2 + S3 + S4 + ...) by (1 - S); this equals (S + S2 + S3 + S4 + ...) - (S2 + S3 + S4 + S5 + ...) = S.)
In the Wizard's case, this is 0.11185 / (1 - 0.11185) = 0.125936.
However, the Wizard is calculating how many free spins you get if your spin generates free spins; this is 10 + 10 S + 10 S2 + S3 + S4 + ... = 10 + 10 x (S + S2 + S3 + S4 + ...) = 10 + 10 x 0.125936 = 11.25936. (I'm not sure where he gets 11.259335.) The 0.125936 number includes the possibility that you don't get any bonus spins.
Let S be the expected number of free spins generated from a particular spin. In the Wizard's example, the probability of generating free spins is 0.011185, and you always get 10 spins, so S = 10 x 0.011185 = 0.11185. Note that if S >= 1, then the expected number of free spins is infinite.
The expected number of free spins from the S spins = S x S.
The expected number of free spins from the S x S spins = (S x S) x S.
And so on.
The total number = S + S2 + S3 + S4 + ...
Since S < 1, this equals S / (1 - S).
(Proof: multiply (S + S2 + S3 + S4 + ...) by (1 - S); this equals (S + S2 + S3 + S4 + ...) - (S2 + S3 + S4 + S5 + ...) = S.)
In the Wizard's case, this is 0.11185 / (1 - 0.11185) = 0.125936.
However, the Wizard is calculating how many free spins you get if your spin generates free spins; this is 10 + 10 S + 10 S2 + S3 + S4 + ... = 10 + 10 x (S + S2 + S3 + S4 + ...) = 10 + 10 x 0.125936 = 11.25936. (I'm not sure where he gets 11.259335.) The 0.125936 number includes the possibility that you don't get any bonus spins.
October 26th, 2015 at 6:59:32 AM
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Thank for your reply and your time.
I guess there is a typo in Wizzard's final value (or a rounding error), because your formula is the same with the Wizzard's in the end. Although it seemed like a geometric series from the begining, could not think of that approach.
I guess there is a typo in Wizzard's final value (or a rounding error), because your formula is the same with the Wizzard's in the end. Although it seemed like a geometric series from the begining, could not think of that approach.
