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.

Wanted to give a little write-up on this wisp because I saw some confusion about it in the PBE thread.

Assuming perfect RNG, the raw expected value is exactly 1 gold, and the wisp itself costs 1 gold:

• 50% → 0g (T)
• 25% → 1g (HT)
• 12.5% → 2g (HHT)
• 6.25% → 3g (HHHT)
• 3.125% → 4g (HHHHT)
• ...

If you average all possible outcomes, you get 1g back on average. So ignoring everything else, buying it has an EV of exactly 0.

Or does it?

Math (sorry)

Let

G = 10n + r,    r ∈ {0, 1, ..., 9}

be your gold when the wisp is offered, and let X be the number of heads before the first tail.

Then:

P(X = k) = 2^-(k+1)

and

E[X] = 1

The wisp costs 1g, so after resolving it your gold is:

G' = G - 1 + X

Now suppose:

1 ≤ r ≤ 9

Buying the wisp cannot knock you down an interest tier in this case, because after paying the 1g you're still above the previous threshold.

To reach the next interest threshold, 10(n + 1), you need:

X ≥ 11 - r

For this geometric distribution:

P(X ≥ m) = 2^-m

So the probability that the wisp earns you at least one extra point of interest is:

2^-(11-r)

For example:

• At 29g, you need X ≥ 2, which happens 25% of the time.
• At 28g, you need X ≥ 3, which happens 12.5% of the time.
• At 21g, you need X ≥ 10, which happens only about 0.01% of the time.

You can theoretically jump multiple interest thresholds with a sufficiently long heads streak, so below the 50g cap the exact additional interest EV is:

EV(G) = Σ[j=0 to 4-n] 2^-(11-r+10j)

for:

r = 1, ..., 9

The later terms are extremely small, though, so for normal gold totals below 50 the result is overwhelmingly determined by:

r = G mod 10

In other words: the last digit of your gold matters a lot.

EV by gold ending

Gold before buying	Approx. Wisp EV
10n + 0	−0.49951g
10n + 1	+0.00098g
10n + 2	+0.00196g
10n + 3	+0.00391g
10n + 4	+0.00782g
10n + 5	+0.01564g
10n + 6	+0.03128g
10n + 7	+0.06256g
10n + 8	+0.12512g
10n + 9	+0.25024g

The wisp's raw gold EV is exactly neutral: you pay 1g and get 1g back on average.

But because TFT interest is based on 10g breakpoints, its actual EV depends heavily on your current gold.

At X9 gold, buying it is quite good: you can't lose an interest tier, and you have a 25% chance to gain one, giving roughly +0.25g of additional EV.

At X0 gold, it is quite bad: spending the 1g immediately drops you an interest tier unless the first flip is heads, giving roughly −0.50g EV.

So:

Buy on X9 > X8 > X7 > ... > avoid on X0.

"But what about hidden intervals?"

Fuck. Okay. The tl;dr above is missing a bit of nuance because it only considers the EV for one round and not multiple rounds.

Oh, and it also doesn't consider the possible +1g from winning, oh dear...

The problem is that in TFT, 1 gold is not always worth exactly 1 gold.

For example, compare 13g and 14g if you're purely just holding:

13 -> 19 -> 25 -> 32 -> 40 -> 49 -> 58

14 -> 20 -> 27 -> 34 -> 42 -> 51 -> 61

That one extra gold at 14 ends up being worth 3 gold by the time both lines reach max interest.

This is basically what people mean by a hidden econ interval: even if 13g and 14g earn the same interest this round, 14g reaches 20 next round while 13g only reaches 19. That changes the next interest payment, which changes the one after that, and so on.

So the wisp's real EV is better thought of as:

EV = average future value after buying - future value without buying

If gold had perfectly linear value, the wisp would still be exactly 0 EV, but sometimes the 1g you risk by buying the wisp is much more valuable than face value.

E.g. your current gold is 14g.

If you don't buy:

14 -> 20

If you buy and immediately flip tails:

14 -> 13 -> 19

You didn't lose interest on the first round, but now you're at 19 instead of 20, so you lose interest on the following round.

Under a deliberately naive model where you never spend, never win, have no streak, and just save until 50, buying the wisp at 14g is actually around -0.9g of eventual EV. So already, the simple rule of "X9 good, X0 bad" starts to fall apart.

There's another problem: win gold happens before interest is calculated.

If you're going to lose the fight:

• Without wisp: stay at 29, get 2 interest
• With wisp: pay to 28
• 0 heads: 28
• 1 head: 29
• 2+ heads: 30+

You have a 25% chance to hit 30, with no chance of losing an interest tier. So:

EV at 29 if you lose ≈ +0.25g

But if you're going to win:

Without the wisp, the +1 win gold already takes you to 30g. With the wisp, you start at 28g, then win gold takes you to 29g.

Now if you flip tails, you've lost the 30g breakpoint. So:

EV at 29 if you win ≈ -0.50g

If your chance of winning the fight is p, the immediate-round EV is approximately:

EV ≈ 0.25 - 0.75p

Which means the wisp at 29g becomes negative EV once your chance to win is above roughly 33%. And that's still only looking at the current round. Once you include hidden intervals, current streak, win probability, and when you're planning to roll down, there isn't really a universal answer based only on your last digit.

The best we can really say in general is:

> The wisp is a 0-EV gamble on the marginal value of your current gold.

Actionable Takeaways

The easiest in-game heuristic is to first estimate your effective gold at the interest check:

• Likely loss: use your current gold
• Likely win: use your current gold + 1
• Unsure: consider both outcomes; if they disagree, err toward skipping

Then look at the last digit of that effective gold:

Effective gold	Wisp
X0	Avoid
X1-X5	Basically neutral; hidden intervals matter more
X6	Slightly +EV
X7	Decent
X8	Good
X9	Best

Or, if you prefer a decision table:

Current gold	Likely loss	Unsure	Likely win
X0	Avoid	Avoid	Basically neutral
X1-X5	Marginal	Marginal	Marginal
X6	Slight +	Slight +	Decent
X7	Decent	Decent	Good
X8	Good	Very good	Best
X9	Best	Avoid	Avoid

At exactly X9, the immediate EV flips negative once you think your chance to win is above about 33%. So if your pool looks remotely even, skip it.

The other important caveat is hidden intervals. The tiny edges at X1-X6 can easily be overwhelmed by whether that 1g changes your econ path next round. I wouldn't try to optimize those from this table, but if you are certain that your game plan is to save to >50g as fast as possible:

Last digit	10s	20s	30s	40s
0	10: -.12	20: -.91	30: -.31	40: -.25
1	11: -.24	21: +.17	31: +.38	41: -.50
2	12: +.52	22: +.35	32: -.25	42: ~0
3	13: +.05	23: -.31	33: -.49	43: ~0
4	14: -.90	24: +.39	34: +.01	44: +.01
5	15: +.19	25: -.23	35: +.02	45: +.02
6	16: +.38	26: -.46	36: +.05	46: +.03
7	17: -.24	27: +.09	37: +.09	47: +.06
8	18: +.52	28: +.17	38: +.19	48: +.12
9	19: +.04	29: +.34	39: +.38	49: +.25

Estimate whether you're winning this fight. If losing, look up your current gold; if winning, look up current gold + 1. Positive = buy, negative = skip. If uncertain, average the two.

So for example:

• 29g, likely loss: look at 29 → +.34g, buy.
• 29g, likely win: look at 30 → -.31g, skip.
• 14g, likely loss: 14 → -.90g, very bad.