Probability of At Least One Win Over Multiple Trials

What This Problem Teaches

Solution: Method 1 — The Complement Approach

The key insight is that "at least one win" is much easier to calculate by thinking about its opposite: "no wins at all."

Step 1 — Convert to loss probability

If Anuhab wins 57% of games, he loses 43% of games:

Step 2 — Express the complement rule

The probability of winning at least one game equals one minus the probability of losing all games:

where n is the number of games played.

Step 3 — Set up the inequality

We need this probability to exceed 99%:

Step 4 — Isolate the exponential term

Rearranging the inequality:

Step 5 — Apply logarithms

Taking the natural logarithm of both sides:

Since ln(0.43) ≈ -0.844 is negative, we flip the inequality when dividing:

Step 6 — Round up to the nearest integer

Since we need a whole number of games, and n must be greater than 5.46:

Solution: Method 2 — Visual Tracking of Losing Streaks

Instead of jumping straight to logarithms, let's build intuition by tracking how the probability of a complete losing streak decreases as we play more games.

Step 1 — Calculate successive losing streak probabilities

Let's compute (0.43)^n for increasing values of n:

Step 2 — Find where we cross the threshold

We need the losing streak probability to drop below 1% (so that the winning probability exceeds 99%). Looking at our calculations:

• After 5 games: P(all losses) = 1.5% > 1%, so P(at least one win) = 98.5% < 99%
• After 6 games: P(all losses) = 0.6% < 1%, so P(at least one win) = 99.4% > 99%

Step 3 — Conclusion

Six games is the minimum needed to guarantee better than 99% chance of at least one win.

Verification

Let's verify our answer by calculating the exact probabilities for 5 and 6 games:

Perfect! With 5 games, Anuhab has a 98.53% chance (not quite 99%), but with 6 games, he has a 99.37% chance, which exceeds our target.

Common Pitfalls

The Underlying Pattern

This problem exemplifies the general formula for "at least one success" problems:

where p is the probability of success on each individual trial.

To find the minimum trials needed for a target probability T:

In our problem: p = 0.57, T = 0.99, so n > ln(0.01) / ln(0.43) = 5.46, requiring 6 games.

Where This Shows Up in Real Life

• Medical testing: How many times should you repeat a test with 85% accuracy to be 99% confident of catching a condition?
• Quality control: If a manufacturing process has a 5% defect rate, how many items should you sample to be 95% sure of finding at least one defect?
• Network reliability: If each server has a 98% uptime, how many backup servers do you need to guarantee 99.9% system availability?

Recognizing This Problem Type

Watch for these telltale phrases that signal a complement probability problem:

• "At least one..." — usually best solved with 1 minus the probability of zero
• "What's the minimum number of trials..." — signals an inequality setup
• "Greater than X% chance" — you'll need to solve an inequality, not just calculate an exact probability
• Individual success probability + multiple trials — geometric probability with the complement rule

The structure is always: individual probability → repeated trials → "at least one" → complement rule → logarithmic inequality.