Average Wrap length when paralysed

[magic9mushroom]

So we all know the formula for Wrap-like moves' effective accuracy when paralysed (normal accuracy * 0.75), but they also have a shorter average length because full paralysis can occur during a Wrap sequence as well (which terminates the sequence). As such, I went and calculated it. Note that the assumption here is that it hits on the first turn; this calculation is solely concerned with mean length of the sequence.

Probabilities:

2 turn Wrap (37.5%)
- 25% 1 turn
- 75% 2 turns

3 turn Wrap (37.5%)
- 25% 1 turn
- 18.75% 2 turns
- 56.25% 3 turns

4 turn Wrap (12.5%)
- 25% 1 turn
- 18.75% 2 turns
- 14.0625% 3 turns
- 42.1875% 4 turns

5 turn Wrap (12.5%)
- 25% 1 turn
- 18.75% 2 turns
- 14.0625% 3 turns
- 10.546875% 4 turns
- 31.640625% 5 turns

Mean = 0.375 * 0.25 * 1 + 0.375 * 0.75 * 2 + 0.375 * 0.25 * 1 + 0.375 * 0.1875 * 2 + 0.375 * 0.5625 * 3 + 0.125 * 0.25 * 1 + 0.125 * 0.1875 * 2 + 0.125 * 0.140625 * 3 + 0.125 * 0.421875 * 4 + 0.125 * 0.25 * 1 + 0.125 * 0.1875 * 2 + 0.125 * 0.140625 * 3 + 0.125 * 0.10546875 * 4 + 0.125 * 0.31640625 * 5

= 2.24658203125

A Wrap-like move hits, on average, around 2.25 times when used by a paralysed Pokemon, instead of its normal average of 3 times.

Just thought that might be nice to know.

[Isa]

So in the end, it's just the average times the chance for a full paralysis? =p

[Crystal_]

It's nice to know the exact numbers. I've always said it's better to Thunder Wave a Dragnite than Blizzard it (unless Blizzard can OHKO). With Wrap's accuracy effectively becoming around 63.5% under paralysis, this goes to show how uneffective wrap moves become when used by a paralyzed Pokemon.

[magic9mushroom]

Jul 1, 2014 2:42:01 GMT -8 Isa said:

So in the end, it's just the average times the chance for a full paralysis? =p

Almost, yes. But I didn't think of that.

[magic9mushroom]

Actually, now that I analyse it more closely, the paralysed Wrap length being the normal length * 0.75 is in fact a coincidence, and not something that could be deduced a priori. If Wrap lasted longer, then the average paralysed length would be shorter than (normal length * 0.75), and vice versa (if it always lasted 2 turns, it'd be an average of 1.75 when paralysed given that it hit, and if it always lasted 100 turns, the average would be far shorter than 75 turns). It just so happens that Wrap's length is such that the average paralysed length is approximately equal to (normal length * 0.75).

[jorgen]

The fact that you didn't get exactly 2.25 (and the error of the final digit was pretty substantial, thereby eliminating machine precision as a possible source of error) made it pretty obvious that a naive a priori deduction was bunk. Still pretty cool that it works out approximately that way though.

I suppose I'd be half-interested to know, for a family of unparalyzed wrap pmfs (perhaps Poisson to make for easy parametric assumptions?), the closed form of the ratio Lbar_par/Lbar as a function of lambda?[1,2,3,...inf), where

Lbar = sum(L*p(n=L|h,ns),L=1:100)/sum(p(n=L|h,ns),L?[1,2,3,...inf));

Lbar_par = sum( L*(0.75^(L-1)*( p(n=L|h,ns) + 0.25*p(n>L|h,ns) ) ),L?[1,2,3,...inf) )/sum( (0.75^(L-1)*( p(n=L|h,ns) + 0.25*p(n>L|h,ns) ),L?[1,2,3,...inf) );

"h" represents "hit", ergo the first hit is assumed a given; and "ns" represents "no switch", ergo the opponent is assumed not to switch out of Wrap.

[magic9mushroom]

Sept 1, 2014 5:09:03 GMT -8 jorgen said:

machine precision as a possible source of error

Those numbers aren't actually rounded at all; that's the full length of the exact mean's decimal expansion (I'd never have quoted so many sig-figs otherwise). All the denominators are powers of 2.

[piexplode]

As a fraction it's a not unpleasant 4601/2048 if you wanted an easier way to use this number in other calculations etc.