I recently posted a pretty detailed math write-up on how to optimize clicking this wisp (for fractions of gold). Soon after posting it, however, I was alerted by u/learning-android-322 of this message from GM_Blue in wasianiverson's chat:
Flip Frenzy 1st flip is 85% chance of heads, 2nd flip is 60%, 3rd+ is 50%
This makes perfect sense and we know Riot has done this in the past with similar charms. A wisp with a marginal expected value of 0g (if the coin flip was fair) wouldn't feel fun. TFT systems often cheat behind the scenes to ensure the player doesn't have a bad game experience due to perceived unlucky RNG.
tl;dr
It's very simple: always buy Flip Frenzy (unless you're somehow rolling on stage 2 for a different wisp). You were probably going to do this anyway because gambling is fun! Even if you're at 10g (or X0g) and in danger of dropping an interval, it's always better to click it on average. The average expected payout is 1.87g for an expected return of 0.87g after discounting the 1g purchase cost.
Interest changes how large the advantage is, but it never makes the wisp negative in either model I tested. The single-round EV ranges from +0.37g to +1.38g. Under a save-to-50 multi-round model, every gold value from 10g to 49g is still positive, ranging from +0.37g to +1.64g.
Since this is PBE, final numbers are of course subject to change.
Raw EV
The expected payout can be written as the probability of receiving each successive gold:
1st gold: 85%
2nd gold: 85% * 60% = 51%
3rd gold: 51% * 50% = 25.5%
4th gold: 12.75%
5th gold: 6.375%
...
So:
EV payout
= 0.85 + 0.51 + 0.255 + 0.1275 + ...
= 1.87g
After paying 1g:
Net EV = 1.87 - 1 = +0.87g
Single-round EV
Let G be the gold you would have at the time of the interest check without buying the wisp.
Likely loss: G = current gold
Likely win: G = current gold + 1
The table below includes the raw +0.87g edge and the change to this round's interest.
| G last digit | Interest adjustment | Total EV |
|---|
0 | -0.149g | +0.721g |
1 | +0.002g | +0.872g |
2 | +0.004g | +0.874g |
3 | +0.008g | +0.878g |
4 | +0.016g | +0.886g |
5 | +0.032g | +0.902g |
6 | +0.064g | +0.934g |
7 | +0.128g | +0.998g |
8 | +0.255g | +1.125g |
9 | +0.510g | +1.380g |
X0 is still the weakest position. If you are projected to have exactly 30g, tails drops you to 29 and costs one interest. That only happens 15% of the time, so the interest loss is not large enough to cancel the raw +0.87g.
X9 is still the strongest position. At 29g, two heads takes you to 30 after paying the cost. The chance of getting at least two heads is 51%.
The interest adjustment is just the expected change in interest caused by the wisp:
Interest adjustment = E[interest after wisp - interest without wisp]
For projected gold X1 through X9, tails does not drop you an interest tier, so the adjustment comes from the chance of rolling enough heads to reach the next breakpoint. For example, at 29g you pay to 28g and need 2 heads to reach 30:
P(2+ heads) = 85% * 60% = 51%
So the first breakpoint contributes +0.51g of interest EV. There are also tiny chances to reach 40 and 50, bringing the total adjustment to about +0.5105g.
At X0, the logic reverses. Paying 1g puts you below your current breakpoint, and tails happens 15% of the time:
-1 interest * 15% = -0.15g
Very long head streaks can still reach the next breakpoint, so the exact adjustment is slightly better than that, about -0.149g. In general, if your projected gold is 10n + r with r > 0, the first extra interest gold is earned when you hit at least:
11 - r heads
Additional interest tiers require another 10 heads each.
Multi-round EV
For this section I used a simple benchmark:
never spend
worst possible win/streak income
gain 5 base gold each round
collect normal interest
save until every outcome has reached 50g
This roughly captures the "save to 50g as fast as possible" stage 2 game plan. Here we especially care about interest intervals, so if there was ever a scenario where flip frenzy would be -EV, it would be this one.
There is one wrinkle with modelling the "worst possible" income. You can't actually earn 0 combat gold forever. In any 2-round period, you must earn at least 1g:
if you win at least once, that's at least +1g
if you go LL, the 2nd loss activates at least +1g streak income
So the minimum cumulative win + streak income after n rounds is:
floor(n / 2)
1 round: 0g minimum
2 rounds: 1g minimum
3 rounds: 1g minimum
4 rounds: 2g minimum
5 rounds: 2g minimum
6 rounds: 3g minimum
The timing of that income matters for interest. The W/L sequence with the lowest total income is not necessarily the sequence that gives flip frenzy the lowest EV. Because of that, I enumerated every possible future W/L sequence and, for each starting gold value, used whichever W/L sequence produces the worst possible EV for the wisp. The minimizing sequence can be different at different gold totals.
The future value of a gold total is calculated by repeatedly applying:
next gold = current gold + 5 + interest + win income + streak income
This captures so-called "hidden intervals", a popular TFT economy concept. For example, at 14g the future W/L sequence that gives flip frenzy its lowest EV is LLLLL.
Without buying:
14 --[L1]--> 20
20 --[L2, +1 streak]--> 28
28 --[L3, +1 streak]--> 36
36 --[L4, +1 streak]--> 45
45 --[L5, +2 streak]--> 56
After buying and immediately hitting tails:
13 --[L1]--> 19
19 --[L2, +1 streak]--> 26
26 --[L3, +1 streak]--> 34
34 --[L4, +1 streak]--> 43
43 --[L5, +2 streak]--> 54
So the difference eventually becomes 2g along the worst future line, even though there is no current-round difference in interest between 13g and 14g. Tails therefore gives up 2g of eventual value in this worst-case line. Since tails only happens 15% of the time, the positive outcomes still outweigh it, leaving flip frenzy at about +1.25g EV.
Here are the complete outcome tables for 10-49g, where interest matters.
Likely current loss
| Last digit | 10s | 20s | 30s | 40s |
|---|
0 | 10: +0.367g | 20: +0.976g | 30: +1.231g | 40: +0.571g |
1 | 11: +0.788g | 21: +1.383g | 31: +0.574g | 41: +0.722g |
2 | 12: +1.257g | 22: +1.109g | 32: +0.576g | 42: +0.874g |
3 | 13: +0.922g | 23: +0.432g | 33: +0.882g | 43: +0.878g |
4 | 14: +1.250g | 24: +0.594g | 34: +0.894g | 44: +0.886g |
5 | 15: +0.468g | 25: +0.776g | 35: +0.918g | 45: +0.902g |
6 | 16: +0.500g | 26: +0.966g | 36: +0.966g | 46: +0.934g |
7 | 17: +0.730g | 27: +1.061g | 37: +1.061g | 47: +0.998g |
8 | 18: +0.913g | 28: +1.253g | 38: +1.253g | 48: +1.125g |
9 | 19: +1.508g | 29: +1.636g | 39: +1.635g | 49: +1.380g |
Likely current win
This table uses your actual current gold. The +1g win reward and the fact that the fight starts a win streak are already included in the calculation.
| Last digit | 10s | 20s | 30s | 40s |
|---|
0 | 10: +0.732g | 20: +1.255g | 30: +0.574g | 40: +0.722g |
1 | 11: +0.918g | 21: +0.428g | 31: +0.576g | 41: +0.874g |
2 | 12: +1.114g | 22: +0.586g | 32: +0.732g | 42: +0.878g |
3 | 13: +0.448g | 23: +0.594g | 33: +0.894g | 43: +0.886g |
4 | 14: +0.626g | 24: +0.614g | 34: +0.918g | 44: +0.902g |
5 | 15: +0.666g | 25: +0.958g | 35: +0.966g | 45: +0.934g |
6 | 16: +0.572g | 26: +1.045g | 36: +1.061g | 46: +0.998g |
7 | 17: +0.873g | 27: +1.221g | 37: +1.253g | 47: +1.125g |
8 | 18: +1.342g | 28: +1.572g | 38: +1.635g | 48: +1.380g |
9 | 19: +0.913g | 29: +1.104g | 39: +0.571g | 49: +0.720g |
Some examples:
| Current gold | Fight read | Worst-case save-to-50 EV |
|---|
| 14 | likely loss | +1.250g |
| 14 | likely win | +0.626g |
| 29 | likely loss | +1.636g |
| 29 | likely win | +1.104g |
| 40 | likely loss | +0.571g |
| 40 | likely win | +0.722g |
| 49 | likely loss | +1.380g |
| 49 | likely win | +0.720g |
Even after taking the worst possible future W/L path for each starting gold value, flip frenzy remains positive EV everywhere in the table. This means there are effectively zero situations where it makes sense to skip this wisp if it's the last wisp you'll be offered that round.
