It has happened to everyone - you think you have the game won, but then an Ice Fang freeze or an untimely critical hit snatches the match right out of your hands. And I am sure that most people have heard their opponent blame "hax" for their loss, and then you smugly think "I would have won anyway."

But how much luck really constitutes hax? Being struck with a critical hit near the end of the game can be crucial, but where is the line that sets apart "bad luck" from "probability"? The purpose of this post is to start a discussion regarding the extent to which we accept the luck factor of the game. When you attack an opponent's pokemon 16 times in a row, do you *expect* to get a critical hit? Would you consider it hax when you do?

When Focus Blast misses, is that considered hax? What about if it hits three times in a row? The probabilities are similar - 30% vs 34.3% - but we intuitively assume that hax only occurs with the most obvious lower probability.

In this thread I will be providing statistics on the chance for a "hax" event to occur with a certain number of attacks. A lot of the chances involved are surprisingly low or high, and it will make you see how incredibly difficult it is to have a hax-free game.

**Case 1: Chance of a Critical Hit with N attacks**

This case involves moves like Return, which have no extra effects. It is the most basic, and the way to calculate it is fairly intuitive. In brief, you take the chance of the event *not* happening, bring it to the Nth power, and subtract that number from 1. That is the probability of an event occurring *at least once* given N iterations. So the chance to critical hit with a single attack is 1 - .9375 ^ 1 (remember, .9375 = 15/16); with two 1 - .9375 ^ 2, and so on.

Code:

Chance of a critical hit occurring with N attacks
1 attack(s)............0.0625
2 attack(s)............0.12109375
3 attack(s)............0.176025390625
4 attack(s)............0.2275238037109375
5 attack(s)............0.2758035659790039
6 attack(s)............0.32106584310531616
7 attack(s)............0.3634992279112339
8 attack(s)............0.4032805261667818
9 attack(s)............0.4405754932813579
10 attack(s)...........0.47553952495127305
11 attack(s)...........0.5083183046418185
12 attack(s)...........0.5390484106017048
13 attack(s)...........0.5678578849390983
14 attack(s)...........0.5948667671304047
15 attack(s)...........0.6201875941847543
16 attack(s)...........0.6439258695482072

This is not particularly impressive; however, the chance to score a critical hit is fairly low. But what happens when we throw additional effects into the mix?

**Case 2: Chance of a Critical Hit OR 10% Effect with N attacks**

This case involves moves like Flamethrower, which has both a 1/16 chance to score a critical hit and a 1/10 chance to Burn. Since the two effects are not mutually exclusive (that is, they occur independently of one another), the chance for one of the events to occur is: (Chance of Crit) + (Chance of Burn) - (Chance of Crit AND Burn) = .15625, which is roughly 15.5%. Now let's pan this out over several attacks, using the same technique as in Case 1:

Code:

Chance of a critical hit OR a 10% effect occurring with N attacks
1 attack(s)............0.15625
2 attack(s)............0.2880859375
3 attack(s)............0.399322509765625
4 attack(s)............0.4931783676147461
5 attack(s)............0.572369247674942
6 attack(s)............0.6391865527257323
7 attack(s)............0.6955636538623367
8 attack(s)............0.7431318329463465
9 attack(s)............0.7832674840484799
10 attack(s)...........0.8171319396659049
11 attack(s)...........0.8457050740931072
12 attack(s)...........0.8698136562660592
13 attack(s)...........0.8901552724744874
14 attack(s)...........0.9073185111503488
15 attack(s)...........0.9217999937831068
16 attack(s)...........0.9340187447544964

Here you can see that the probability of a "hax" event occurring when using this type of attack increases very quickly! In fact, by the time the fourth attack is used, the chance of a player getting haxed is very nearly 50%. This means that if each player uses Flamethrower twice in a row, there is about a coin flip's chance that at least one player will be hit with either a Burn or a Critical hit. This same formula can be used to calculate the chance of a hax with a 20% effect rate (such as Waterfall or Crunch) or a 30% effect rate (Such as Iron Head or Lava Plume).

Code:

Chance of a critical hit OR a 20% effect with N attacks
1 attack(s)............0.25
2 attack(s)............0.4375
3 attack(s)............0.578125
4 attack(s)............0.68359375
5 attack(s)............0.7626953125
6 attack(s)............0.822021484375
7 attack(s)............0.86651611328125
8 attack(s)............0.8998870849609375
9 attack(s)............0.9249153137207031
10 attack(s)...........0.9436864852905273
11 attack(s)...........0.9577648639678955
12 attack(s)...........0.9683236479759216

Perhaps this is what makes Gyarados so popular? With just two attacks, the chance to score a flinch or crit is greater than the chance of hypnosis missing; with 3, the chance is almost that of hypnosis *hitting*, and with four the chance to achieve a hax event is almost 70%.

Code:

Chance of a critical hit OR a 30% effect with N attacks
1 attack(s)............0.34375
2 attack(s)............0.5693359375
3 attack(s)............0.717376708984375
4 attack(s)............0.8145284652709961
5 attack(s)............0.8782843053340912
6 attack(s)............0.9201240753754973
7 attack(s)............0.9475814244651701
8 attack(s)............0.9656003098052679
9 attack(s)............0.9774252033097071
...

and just for kicks...

Code:

Chance of Serene Grace Air Slash flinching N times consecutively
1 attack(s)............0.57
2 attack(s)............0.32489999999999997
3 attack(s)............0.18519299999999994
4 attack(s)............0.10556000999999997
5 attack(s)............0.06016920569999997
6 attack(s)............0.03429644724899998
7 attack(s)............0.019548974931929988
8 attack(s)............0.011142915711200092
...

So now you can justify your whining when you get flinched four times in a row by togekiss; the chance is the same as scoring a burn with Flamethrower!

**Case 3: Chance of a Critical Hit OR Multiple Effects with N attacks**

This case really only applies to the "fang" attacks, which have a 10% chance to either Flinch or Freeze. What complicates this even more is the fact that a frozen foe has a chance to thaw on the first turn (I am using 60 / 256, roughly 23.4%). So let's find the chance to score a flinch OR freeze OR crit, AND NOT have the foe thaw the turn they are frozen, AND have the attack hit:
- First we will find the chance to freeze OR flinch: .1 + .1 - (.1 * .1) = .19
- Next, we will find the chance of this OR a critical hit occurring: .19 + .0625 - (.19 * .0625) = .240625
- Then muliply this chance by the probability of NOT thawing and the probability of the attack hitting: .240625 * (1 - .234375) * .95 = .175017 or about 17.5%.

Of course, the chance for fire fang and thunder fang is slightly higher, since their statuses never "go away". However Ice Fang is probably used much more than both of those combined, so I felt it made more sense to show the calculations for that one.

Code:

Chance of Ice Fance achieving either a critical hit, flinch, or a freeze, and the foe NOT defrosting the first turn, and the attack hitting
1 attack(s)............0.17501708979999997
2 attack(s)............0.3194031978779387
3 attack(s)............0.4385192695125283
4 attack(s)............0.5367879929412237
5 attack(s)............0.6178580103770678
6 attack(s)............0.6847393892912552
7 attack(s)............0.7399153839060704
8 attack(s)............0.7854346365165802
9 attack(s)............0.8229872420053275
10 attack(s)...........0.8539674997670268
11 attack(s)...........0.8795256829740196
12 attack(s)...........0.9006107473355492
13 attack(s)...........0.9180055650942783
14 attack(s)...........0.9323559924712732
15 attack(s)...........0.9441948498113603
16 attack(s)...........0.953961704793228

So if you need a hax event with Gyarados, Waterfall is the way to go (even though it may not seem like it at first).

Here are a few more I found interesting:

Code:

Chance of a Hax event with N attacks of Thunder
1 attack(s)............0.24062499999999998
2 attack(s)............0.423349609375
3 attack(s)............0.5621061096191406
4 attack(s)............0.6674743269920349
5 attack(s)............0.7474883170595765
6 attack(s)............0.8082489407671158
7 attack(s)............0.8543890393950286
8 attack(s)............0.8894266767905998
9 attack(s)............0.9160333826878617
10 attack(s)...........0.936237849978595
11 attack(s)...........0.9515806173274955
12 attack(s)...........0.9632315312830669
13 attack(s)...........0.9720789440680789
14 attack(s)...........0.9787974481516974
15 attack(s)...........0.9838993121901952
16 attack(s)...........0.9877735401944295

Perhaps Thunder could be more useful on a stall team, which frequently has turns to spare? Of course, you would likely need a reliable way of deal with Gyarados!

Code:

Chance of Focus Blast hitting N times consecutively.
1 attack(s)............0.7
2 attack(s)............0.49
3 attack(s)............0.343
4 attack(s)............0.2401
5 attack(s)............0.16807
6 attack(s)............0.117649
7 attack(s)............0.0823543
8 attack(s)............0.05764801

I hope this helps to redefine your idea of "hax". In a single battle, it only makes sense that several hax events would occur, given the number of attacks in a single battle. In fact, *not* having any of these events occur could be considered hax, since the chance of that happening is obviously very low.

So the next time you are Calm Minding your Suicune against a Magnezone, do not get upset when you get critted or paralyzed by its fourth Thunderbolt!