Answer
Well, I guess no one was interested, but here is the answer anyway. If the probability of a strike is x, the probability of a spare is s, and the first ball in a frame gets nine pins if it is not a strike, then the score in a given frame is:
30 for three strikes
29 for two strikes followed by a nonstrike
20 for a strike followed by a spare
19 for a strike followed by an open frame
20 for a spare followed by a strike
19 for a spare followed by a nonstrike
9 for an open frame
Converting each possibility to an expected value, we get
30*x*x*x +
29*x*x*(1-x) +
20*x*s +
19*x*(1-x-s) +
20*s*x
19*s*(1-x) +
9*(1-x-s).
That simplifies to
9 + 10*s + 10*x + 2*s*x + 10*x^2 + x^3.
So for a full game the average score would be
90 + 100*s + 100*x + 20*s*x + 100*x^2 + 10*x^3.
I have not been able to figure out the standard deviation without simulation. For a typical 140 average bowler, I get a standard deviation of about 23.
Any comments? Does the formula look reasonable?
Cheers,
Irchans
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