Other games

Conclusion: either I am statistically incredibly unlucky, or the devs are lying to us about the rates. I present the data for you to decide for yourself.

There is constant ongoing debate about whether or not the posted pull rate includes pity or not. asked the support chat, and received this response in the last day or so:

However, many of us feel like this can't possibly be true because the number of times we go to pity, or get a single A-tier hero on a 10-pull, seems much greater than is explicable if this were the case.

So I've done the analysis to see how I compare to this supposed clarification from the devs.

I've have been taking screenshots of every pull for some time. I have done a careful check, through my archive and I have 550 consecutive 10-pulls on the standard banner. This is enough for some fairly robust analysis.

From these pulls I received 12 S-tier heroes and 115 A-tier heroes. This works out to:

• 2.18% chance of S-tier hero

• 20.91% chance of A-tier hero

These rates are close to the posted rates of:

• 2.05% chance of S-tier hero

• 22.5% chance of A-tier hero

This is about what we would expect if the rates included pity. However, the devs claim that the rates do not include pity. What should my expected number of heroes have been if that were true?

I created a model with the following assumptions:

• Each pull has a 22.5% chance of pulling an A-tier hero and a 2.05% chance of pulling an S-tier hero.

• Pulling an A-tier hero and S-tier hero are mutually exclusive (obviously)

• Guaranteed S-tier hero on the 60th pull with no S-tier

• Guaranteed A-tier hero on the 10th pull with no card (either S-tier or A-tier)

Why did I make this last assumption? Because of situations like this:

I ran this model 10,000 times, simulating 550 pulls each time for a total of 5.5 million pulls. These were the results:

The actual rates yielded from this simulation over all 5.5 million pulls were:

• 2.81% chance of S-tier hero

• 23.93% chance of A-tier hero

These would be close to the 'actual' rates if the devs statement on pity is true.

The mean number of expected number of heroes was:

• 15.45 S-tier heroes (I actually got 12)

• 131.60 S-tier heroes (I actually got 115)

When compared against these results, I am in the bottom 10.68% for S-tier pulls, and in the bottom 2.4% for A-tier pulls.

I also checked how many times I drew a single a-tier hero (referred to here as the A-tier pity).

If there's really a 22.5% chance for an A-tier hero with each pull on this banner, you should expect a 0.775^9 = 10.09% chance for any given 10-pull to yield a single A-tier hero. I hit this 16 times, or 29.09% of the time - 30.91% of the time if you count the single S-tier pull shown above. This is over three times the expected rate!

Simulating 5.5 million pulls, the odds of getting 16 or more A-tier pities is 4.04%; or just 1.86% for 17 or more when you include the single S-tier pull shown above.

Now it's possible that I am just in the unluckiest 10.68% of the game for S-tier pulls. But to ALSO be in the unluckiest 2.4% for A-tier pulls, AND 1.86% for 10-pull pities, starts to strain credibility somewhat. Now, A-tier pities are related to A-tier draws, so they are not fully independent. But the S-tier draws and A-tier draws are independent, so we can simply multiply the probably together to get my total luck.

If we believe the devs that the posted rates do not include pity, I am in the bottom 0.26% of all players for luck. To be precise, only one in every 390 players is unlucky as me.

What most strains credibility is I'm top 100 on my server - which has about 50-60 whales (or at least chonkier dolphins). No-one in my guild with a similar spend to me appears to have the *expected* number of heroes indicated by this analysis, and many have less (I bought the first $1 bundle and the Noble pass only). If I'm THIS unlucky, but also top 100 and/or on par with everyone else, then I'm either a genius or everyone else is equally unlucky.

I don't think I'm a genius.

I think the devs are deliberately misleading us.

Is this data enough to be absolutely certain? No. But it looks bad.

Over longer timeframes, I should experience regression towards the mean if what the devs are claiming is true. I'm going to keep monitoring, and I'll update when I have 1000 pulls (so... months from now, probably, unless someone wants to front me the cash for growth bundles).

If anyone else wants to keep their own tally of pulls (or has been keeping one) feel free to PM me with details for analysis.