Today, we do statistics 101, where we explain how basic probability works on Bernoulli trials with multiple draws
When you do any pull on Atlantis, you have a single pull 2.9% chance of getting an Atlantis 5*, Featured Past HOTM, or Current HOTM (.3% + 1.3% + 1.3%). We will call this a Success on a pull, for convenience.
1 draw
On a single draw, the outcomes are easy:
2.9% chance of Success, and 97.1% chance of Failure.
Ok, easy enough. But what happens when we do two draws?
2 draws
The possible outcomes of 2 draws are:
| 1st Draw | 2nd Draw |
|---|---|
| Failure | Failure |
| Failure | Success |
| Success | Failure |
| Success | Success |
So, for instance, on the first line, we failed to get one of our target heroes on the first draw, then failed again in the second. That’s one possible outcome.
The second line represents another possible outcome: we failed to get one of our target heroes on the first draw, but succeeded on the second draw.
Each draw is independent, so they have the same individual probability of success. But we multiply probabilities to aggregate them. So, here are the probabilities of the possible outcomes:
| 1st Draw | 2nd Draw | Aggregate Probability |
|---|---|---|
| 97.1% | 97.1% | 94.3% |
| 97.1% | 2.9% | 2.8% |
| 2.9% | 97.1% | 2.8% |
| 2.9% | 2.9% | 0.1% |
| 100.0% |
We can see that the total probability sums down the last column to 100%.
Now, let’s sum up the aggregate probability of the cases where we did succeed, the second, third and fourth cases:
Probability of success in 2 draws = 2.8%+2.8%+0.1%=5.7%
5.7% of the time when we do 2 draws, we will Succeed (5* Atlantis, Featured Past HOTM, or Current HOTM).
Ok, what about 3 draws?
3 Draws
The outcomes are:
| 1st Draw | 2nd Draw | 3rd Draw |
|---|---|---|
| Failure | Failure | Failure |
| Failure | Failure | Success |
| Failure | Success | Failure |
| Failure | Success | Success |
| Success | Failure | Failure |
| Success | Failure | Success |
| Success | Success | Failure |
| Success | Success | Success |
Now, let’s apply the single trial probabilities and see what our aggregate chances of success are:
| 1st Draw | 2nd Draw | 3rd Draw | Aggregate Probability |
|---|---|---|---|
| 97.1% | 97.1% | 97.1% | 91.55% |
| 97.1% | 97.1% | 2.9% | 2.73% |
| 97.1% | 2.9% | 97.1% | 2.73% |
| 97.1% | 2.9% | 2.9% | 0.08% |
| 2.9% | 97.1% | 97.1% | 2.73% |
| 2.9% | 97.1% | 2.9% | 0.08% |
| 2.9% | 2.9% | 97.1% | 0.08% |
| 2.9% | 2.9% | 2.9% | 0.00% |
| 100.00% |
Now, let’s sum up the aggregate probability of all the cases where we did succeed at least once: the 2nd through 8th cases:
Probability of success in 3 draws = 2.73%+2.73%+0.08%+2.73%+0.08%+0.08%=8.43%
8.43% of the time when we do 3 draws, we will Succeed (5* Atlantis, Featured Past HOTM, or Current HOTM).
What does this mean for multiple draws?
Well, we can see that the probability of getting a successful outcome grows with each additional draw we do:
1 draw: 2.9%
2 draws: 5.7%
3 draws: 8.43%
Indeed, if we kept doing this, our aggregate probability of a successful outcome would climb to 1 as the number of draws became infinite.
How to calculate the probability of at least one success
We can see from our tables that only the first line represents complete failure.
1 draw: 97.1%
2 draws: 97.1% * 97 1% = 94.3%
3 draws: 9.71% * 97.1% * 97.1% = 91.55%
The pattern is pretty easy to see. The probability of complete failure is 97.1%^(# of draws)
The total probability across all outcomes must sum to 100%. So, if we know the probability of complete failure, then the probability of at least some success is:
Probability of Success = 100% - 97.1%^(# of draws)