Maths
Fuse odds explained: how many tries do you need?
If one fusion gives your target pet with chance p, how many fusions until you probably get it? The answer only depends on p, and the tables below are computed with the same code as the calculator.
Facts checked . The game updates often, so check your own fusion menu before you spend anything. Unofficial fan site. The maths here is exact, but whether the game fits the model is not known.
The formula
Suppose every fusion is independent and has the same chance p of giving the pet you want. After n fusions:
- Chance of no hit: (1 − p)n
- Chance of at least one hit: 1 − (1 − p)n
- Fusions needed for a chance c: the smallest whole n with 1 − (1 − p)n ≥ c
Worked example from a reported preview: the fourth result had a 3% chance, each fusion cost 1B coins and took 30 minutes. With a 10B budget you get 10 fusions, and the chance of at least one hit is 1 − 0.9710 ≈ 26.26%. To reach 90% you need 76 fusions: 76B coins and 38 hours of machine time.
Fusions needed for a chance of at least one hit
| Chance per fusion | 50% sure | 80% sure | 90% sure | 95% sure |
|---|---|---|---|---|
| 0.5% | 139 | 322 | 460 | 598 |
| 1% | 69 | 161 | 230 | 299 |
| 2% | 35 | 80 | 114 | 149 |
| 3% | 23 | 53 | 76 | 99 |
| 5% | 14 | 32 | 45 | 59 |
| 10% | 7 | 16 | 22 | 29 |
| 15% | 5 | 10 | 15 | 19 |
| 25% | 3 | 6 | 9 | 11 |
| 50% | 1 | 3 | 4 | 5 |
Multiply any entry by the cost of one fusion for the coins, and by the time of one fusion for the machine time. 76 fusions at 1B and 30 minutes is 76B and 38 hours.
Chance of at least one hit after a fixed number of fusions
| Chance per fusion | 1 fusion | 5 fusions | 10 fusions | 25 fusions | 50 fusions | 100 fusions |
|---|---|---|---|---|---|---|
| 0.5% | 0.50% | 2.48% | 4.89% | 11.78% | 22.17% | 39.42% |
| 1% | 1.00% | 4.90% | 9.56% | 22.22% | 39.50% | 63.40% |
| 2% | 2.00% | 9.61% | 18.29% | 39.65% | 63.58% | 86.74% |
| 3% | 3.00% | 14.13% | 26.26% | 53.30% | 78.19% | 95.24% |
| 5% | 5.00% | 22.62% | 40.13% | 72.26% | 92.31% | 99.41% |
| 10% | 10.00% | 40.95% | 65.13% | 92.82% | 99.48% | >99.99% |
| 15% | 15.00% | 55.63% | 80.31% | 98.28% | 99.97% | >99.99% |
| 25% | 25.00% | 76.27% | 94.37% | 99.92% | >99.99% | >99.99% |
| 50% | 50.00% | 96.88% | 99.90% | >99.99% | >99.99% | 100% |
An average is not a promise
At 3% the average is about 33 fusions per hit (1 ÷ 0.03). That does not mean 33 fusions will do it. After 33 fusions the chance of having hit at least once is about 63%, so about one player in three will still be empty-handed. And because every fusion is a fresh roll in this model, a long miss streak does not make the next one more likely.
What the model leaves out
- Independence. If the game had streak protection, odds would improve with every miss. Nothing we read says it does or does not.
- Changing chances. The preview depends on the four pets you deposit. A different set of pets is a different roll, so only reuse the same preview for the same four pets.
- The pets you spend. Cost here is coins and time. Four pets are used up in each fusion, and that cost is yours to weigh.
- More than one prize. If you would be happy with any of several results, add their chances together and enter the sum as the target.
Plug in your own numbers in the fuse calculator.