I Flipped 3 Billion Coins So You Don't Have To

My heads chance was sitting at 55%. I'd been clicking for what felt like forever, watching my streak counter reset over and over. "How much longer is this gonna take?" I muttered, staring at my measly 7-combo that just got wiped by a single tails.

So I did what any reasonable person would do: I wrote a simulator in Rust and flipped 3 billion virtual coins to figure out exactly how fucked I was.

I found Unfair Flips through a YouTube video where some guy spent way too long flipping coins and slowly losing his mind. Watching him suffer through the probability was hilarious enough that I bought the game myself for $1.99. The premise sounded simple enough: flip coins until you get 10 heads in a row. Your starting coin is weighted so badly that heads only comes up 20% of the time. You can buy upgrades to improve your odds, but the game doesn't tell you how much they actually help. It just shows you your current win chance and a streak counter that resets every time you hit tails. So you're sitting there clicking, watching numbers go up and down, with no idea if you're close to winning or if you're going to be here for another hour. That lack of information drove me crazy enough to run the actual numbers.

The Setup

In Unfair Flips, you start with a pathetic 20% chance of heads and need to get 10 consecutive heads to win. Each flip takes 2 seconds. You can upgrade your head probability by 5% increments up to 60%. Sounds simple. It's not.

Here's how long you'll be clicking at each upgrade level:

Head Chance	Flips to Win	Real Time
20% (start)	11,511,628	266 days
25%	1,287,695	30 days
30%	257,589	6 days
35%	55,159	31 hours
40%	16,170	9 hours
45%	5,322	3 hours
50%	2,054	68 minutes
55%	875	29 minutes
60% (max)	411	14 minutes

At 20%, you're looking at clicking a button for 266 straight days, and the math isn't wrong.

Your first upgrade matters more than you'd think. It's the difference between "I'll finish this eventually" and "I should write this into my will." Here's what each +5% upgrade buys you:

Upgrade	Speedup
20% → 25%	8.94x faster
25% → 30%	5.00x faster
30% → 35%	4.67x faster
35% → 40%	3.41x faster
40% → 45%	3.04x faster
45% → 50%	2.59x faster
50% → 55%	2.35x faster
55% → 60%	2.13x faster

That first upgrade is nearly 9x faster. The value of upgrades decreases as you go higher, but they're all worth it. Going from 20% to 60% makes you 28,034 times faster, which is like comparing walking to teleportation.

The Math Behind the Misery

Here's the formula for expected flips to get $k$ consecutive heads with probability $p$:

$$E[k] = \frac{1}{p} + \left(\frac{1}{p}\right)^2 + \left(\frac{1}{p}\right)^3 + \cdots + \left(\frac{1}{p}\right)^k$$

For 10 consecutive heads at 20%, you're looking at $\frac{1}{p} = 5$, which means $E[10] = 5 + 25 + 125 + \cdots + 5^{10} \approx$ 12.2 million flips. At 50%, it's $\frac{1}{p} = 2$, so $E[10] = 2 + 4 + 8 + \cdots + 1024 \approx$ 2,046 flips. Every percentage point matters because you're raising numbers to the 10th power. Small changes compound into massive differences that will make you question your life choices.

Common misconception time: "The probability of 10 heads is $(0.5)^{10} = 0.098\%$, so I should see it once every ~1000 flips." Wrong. That's the probability of getting 10 heads on your next 10 flips specifically. We're asking a completely different question: "How many flips until I first see a streak of 10?" That involves expected waiting times and Markov chains and all sorts of probability theory that makes your head hurt.

At 50%, the answer isn't 1,024 flips. It's 2,046. Almost exactly double. The extra flips account for all the failed attempts where you get 7, 8, 9 heads then hit tails and have to start over. You know, that thing that happens constantly and makes you want to throw your mouse through the monitor.

Even if you're not shooting for 10, combos still matter for points. Here's what to expect:

Combo	20%	40%	60%
3x	145 flips (5 min)	10 flips (20 sec)	5 flips (10 sec)
5x	3,905 flips (2.2 hr)	78 flips (2.6 min)	20 flips (40 sec)
7x	97,655 flips (2.3 days)	609 flips (20 min)	77 flips (2.6 min)
10x	12.2M flips (282 days)	16,170 flips (9 hr)	411 flips (14 min)

At 20%, even a 5x combo takes over 2 hours on average. At 60%, you'll see 7x combos every few minutes. This is why starting the game feels like punishment.

Optimal Strategy

Based on the math, rush probability upgrades. Nothing else matters until you hit at least 40%. Don't expect combos early. At 20-30%, you'll spend most of your time seeing 1s, 2s, and 3s. A 5x combo is a rare celebration worth screenshotting because you won't see another one for hours.

The 40% threshold is where the game becomes "playable." 9 hours to win is long but not insane. 50% is the sweet spot, just over an hour to win. Most runs should aim here before pushing for 60%. And 60% is luxury. Nice to have, but the jump from 55% to 60% only saves you 15 minutes. Earlier upgrades are way more impactful.

What Theory Can't Tell You

The theoretical formula gives you the average. But in reality, you might be way luckier or way unluckier than average. At 20% chance, for 10 consecutive heads, the mean is 11.5 million flips but the standard deviation is 13.8 million flips. The standard deviation is larger than the mean itself. That means massive variance. Some runs take over 6x longer than average. You could be the unlucky soul.

Probability	Avg Flips	Luckiest Run	Unluckiest Run	Variance Factor
20%	11.5M	241k (48x faster)	75.8M (6.6x slower)	315x spread
50%	2,054	173 (12x faster)	17,393 (8.5x slower)	100x spread
60%	411	26 (16x faster)	3,445 (8.4x slower)	132x spread

At 20%, the luckiest person finished in 241,055 flips while the unluckiest needed 75,821,268 flips. That's a 315x difference between best and worst case. Imagine being that unlucky person. Imagine the despair.

The median vs mean situation is also brutal. At 20%, the mean is 11.5M flips (266 days) but the median is 6.7M flips (155 days). The median is 42% lower than the mean. This distribution is heavily right-skewed. Half of players will finish faster than 155 days, but some unlucky souls drag the average way up. You'll probably beat the average. But if you're unlucky, you'll be waiting a LONG time while questioning every decision that led you to this moment.

The coefficient of variation (std/mean) tells us how "frustrating" each probability level is:

Probability	CV (Std/Mean)
20%	1.20 (120%)
40%	0.98 (98%)
60%	1.00 (100%)

A CV over 100% means the standard deviation exceeds the mean. All probability levels have extremely high variance. You cannot predict how long you'll wait. The theoretical average is just a rough guide. For all you know, you're the outlier who gets to suffer extra.

Because the distribution is geometric-like, your expected remaining time is always the same, no matter how long you've been trying. If you've been clicking for 30 minutes at 50% and haven't won yet, your expected time to win is still ~68 minutes. Your past failures don't bring you any closer to success. This is the mathematically justified version of "this slot machine is due to pay out!" Except it's not due. It was never due. You've been lied to by probability.

The spread between luckiest and unluckiest runs is massive at every probability level. Someone's getting lucky while you suffer.

Coefficient of variation over 100% means the standard deviation exceeds the average, which makes this extremely unpredictable and pretty much hell.

The mean is consistently higher than the median, meaning unlucky outliers drag up the average. You might be that outlier.

Early upgrades give huge speedups; later ones help but matter less. Get to 40% before you do anything else.

From 20% to 60% you get 28,000x faster, and this chart is the only reason I kept playing.

How I Computed This

I wrote a Rust program that simulates flipping until 10 consecutive heads, records when each streak (1-10) first occurs, runs thousands of trials per probability level, and uses parallel processing across all CPU cores. Performance stats: 3 billion coin flips simulated, 1.03 billion flips per second, 3 seconds total computation time. Lower probabilities need fewer trials because each trial takes millions of flips. At 20%, each simulated "game" required ~12 million flips on average.

The results match theoretical predictions within a few percent, confirming both the math and the simulation. Which means all of this is real. None of this is a mistake. The suffering is intentional.

The Verdict

If you're stuck at 55% wondering how long until you win: 29 minutes average. But that's just the average. You might get lucky in 5 minutes. You might also go an hour. Probability is cruel like that. At least now you know: you're not unlucky. You're just experiencing exactly what the math predicted.

And if you're still at 20%, put the game down and go outside. 266 days is not a typo, it's a warning.