Probability of at Least One Success Calculator

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:

1. Find the probability of failure in a single trial, which is \( 1 - P \).
2. Raise the failure probability to the power of \( n \) (the number of trials) to find the probability of no successes in \( n \) trials.
3. 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.