Calculate the probability of achieving at least one success given the probability of success in a single trial and the total number of trials.

## Understanding the Probability of At Least One Success

In probability, an **independent event** is one where the outcome of one event does not affect the outcome of another. For example, when flipping a coin twice, the outcome of the first flip does not affect the outcome of the second flip.

### Formula

The probability of at least one success in a series of \( n \) independent trials, each with a probability of success \( P \) in a single trial, is calculated by finding the probability of failure in all trials and subtracting it from 1:

$$ P(\text{at least one success}) = 1 - (1 - P)^n $$

Here:

- \( P \): Probability of success in a single trial
- \( n \): Number of independent trials

### How to Calculate

To calculate the probability of at least one success, follow these steps:

- Find the probability of failure in a single trial, which is \( 1 - P \).
- Raise the failure probability to the power of \( n \) (the number of trials) to find the probability of no successes in \( n \) trials.
- Subtract this result from 1 to find the probability of at least one success.

### Example

Suppose you have a game where the probability of winning (success) in a single round is 0.2 (or 20%). If you play this game 5 times, what is the probability of winning at least once?

Using the formula:

$$ P(\text{at least one success}) = 1 - (1 - 0.2)^5 $$

Calculating this:

- The probability of failure in a single trial is \( 1 - 0.2 = 0.8 \).
- Then \( 0.8^5 \approx 0.32768 \).
- So, \( P(\text{at least one success}) = 1 - 0.32768 = 0.67232 \) or about 67.2%.

This means there is approximately a 67.2% chance of winning at least once if you play the game 5 times.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.