1/5 X 1/5 X 1/5 X 1/5

You're staring at a fraction multiplied by itself four times. Now, maybe it showed up on a homework assignment. So maybe you're calculating odds for a game. Maybe you just typed it into a search bar out of curiosity.

Here's the short answer: 1/625. Or 0.0016 if you prefer decimals.

But the real answer? That depends entirely on why you're asking.

What Is 1/5 × 1/5 × 1/5 × 1/5

At its core, this is exponentiation in disguise. You're raising one-fifth to the fourth power:

(1/5)⁴

The numerator stays 1 (because 1 × 1 × 1 × 1 = 1). The denominator becomes 5⁴, which is 5 × 5 × 5 × 5 = 625.

So the exact fraction is 1/625.

As a decimal, that's 0.0016. As a percentage, 0.16%.

The Exponent Shortcut

If you see a fraction multiplied by itself repeatedly, you're looking at a power. The general rule:

(a/b)ⁿ = aⁿ / bⁿ

In this case: (1/5)⁴ = 1⁴ / 5⁴ = 1/625.

This works for any fraction raised to any positive integer power. That said, negative powers flip the fraction first. Fractional powers introduce roots. But for whole-number exponents? Just multiply the top by itself n times, same for the bottom.

Why the Denominator Explodes

Here's what trips people up: the denominator grows fast*.

• 1/5 = 0.2 (20%)
• (1/5)² = 1/25 = 0.04 (4%)
• (1/5)³ = 1/125 = 0.008 (0.8%)
• (1/5)⁴ = 1/625 = 0.0016 (0.16%)

Each multiplication by 1/5 cuts the value to one-fifth of what it was. Four rounds of that and you're down to a sliver.

Why It Matters / Why People Care

You might wonder: who actually multiplies 1/5 by itself four times outside of a math textbook?*

More people than you'd think.

Probability and Independent Events

This is the big one. 1/5 × 1/5 × 1/5 × 1/5 is the probability of four independent events each* happening, when each has a 1/5 (20%) chance.

Real examples:

• Drawing a specific card from a 5-card deck, replacing it, and doing that four times in a row
• Rolling a specific number on a 5-sided die (or a 10-sided die where two faces count) four times straight
• A 20% chance of rain each day for four days — the chance it rains all four days*
• A game mechanic with a 20% proc rate triggering on four consecutive hits

The key word is independent. Even so, the outcome of one doesn't affect the others. If you're drawing cards without* replacement, the math changes completely.

Compounding Decay

Anything that loses 80% of its value repeatedly follows this pattern.

• A radioactive isotope with a "fifth-life" (hypothetical) instead of half-life
• A chemical reaction where only 20% of reactant survives each stage
• Signal attenuation through four identical filters that each pass 20%
• Money losing purchasing power if inflation somehow hit 80% per period (extreme, but mathematically identical)

Gaming and Loot Tables

Gamers run into this constantly.

World of Warcraft* mount drops. Destiny* exotic engrams. Genshin Impact* 5-star pulls (though those have pity systems that break independence). Any "1 in 5" chance repeated four times.

Players feel* like four tries at 20% should guarantee success. Now, the math says: nope, you still have a 59% chance of zero successes. We'll get to that.

How It Works (and How to Calculate It)

By Hand

Method 1: Multiply straight across

1/5 × 1/5 = 1/25
1/25 × 1/5 = 1/125
1/125 × 1/5 = 1/625

Method 2: Exponent rules

For more on this topic, read our article on how many hours is 1000 minutes or check out consider the following three systems of linear equations.

(1/5)⁴ = 1⁴ / 5⁴ = 1 / (5×5×5×5) = 1/625

Method 3: Decimal conversion

0.2 × 0.2 = 0.04
0.04 × 0.2 = 0.008
0.008 × 0.2 = 0.0016

All three give the same result. Method 2 is fastest once you're comfortable with exponents.

On a Calculator

• Scientific calculator: Type 1 ÷ 5 ^ 4 = or use the x^y button: ( 1 ÷ 5 ) ^ 4 =
• Phone calculator (landscape mode on iOS/Android): same as scientific
• Google search bar: type (1/5)^4 or 1/5*1/5*1/5*1/5
• Excel/Sheets: =(1/5)^4 or =POWER(1/5,4)

In Code

# Python
result = (1/5) ** 4
# 0.0016

# JavaScript
const result = Math.pow(1/5, 4);
// 0.0016

# R
result <- (1/5)^4
# 0.0016

Floating point precision might show 0.0016000000000000001 in some languages. And that's normal. Round for display.

The "At Least Once" Flip

Here's what most people actually* want when they're asking about repeated 20% chances: what's the probability it happens at least once in four tries?*

That's 1 minus the probability it never happens.

• Chance of failure on one try: 4/5 = 0.8
• Chance of failure all four times: (4/5)⁴ = 256/625 = 0.4096
• Chance of at least one success*: 1 - 0.4096 = 0.5904 (59.04%)

So four independent 20% chances gives you roughly a 59% shot at seeing it happen at least once*. And not guaranteed. Not even 2/3.

Expected Value

If you repeat this 4-trial sequence many times, the average number of successes per sequence is:

4 × (1/5) = 0.8

Less than one. Over 100 sequences (400 total trials), you'd expect ~80 successes. But any single sequence of four?

0.96% chance). That's the most common result. One success happens 40.96% of the time. Two or more? Only 18.08%.

This is why slot machines and loot boxes are so profitable. The math is designed so that the most frequent outcome is the one that pays out the least.

The Gambler's Fallacy

This entire scenario is the mathematical heart of the gambler's fallacy: the belief that if something happens more frequently than normal during a given period, it will happen less frequently in the future, or vice versa.

A player who has had four failed 20% pulls might think, "I'm due for a win!Because of that, each pull is an independent event. On the flip side, the machine has no memory. " They are not. The odds reset to 1/5 every single time.

The only exception in gaming is explicit "pity systems," like Genshin Impact*'s, which guarantee a 5-star character after a certain number of pulls. This is a deliberate break from pure probability to improve player experience. It's a concession that the pure math is too punishing.

Beyond Gaming: Real-World Consequences

This isn't just abstract. This pattern of repeated small chances appears everywhere:

• Medicine: A treatment with a 20% success rate per dose. After four doses, the chance of at least one* success is still under 60%. Patients may lose hope, but the math explains why.
• Sales: A salesperson with a 20% close rate on calls. After four calls, they are more likely to have made zero sales than one. Their day is likely to be frustrating.
• Project Management: A task with a 20% chance of being completed on time each week. After four weeks, there's a 41% chance it was never completed on time in any given week.

Conclusion

The journey from a single 1-in-5 chance to a 1-in-625 chance after four attempts is a stark lesson in exponential decay. It reveals a fundamental mismatch between our intuition and statistical reality. Here's the thing — we expect linear thinking—four times the attempts should mean four times the odds—but probability compounds multiplicatively, creating outcomes that feel far more rare than we anticipate. Understanding this principle is not just about calculating numbers; it's about recognizing the hidden structure of risk and reward in games, business, and life, and guarding against the seductive but false logic of the gambler's fallacy.