Quick Summary

The Binomial Distribution calculates the probability of getting a specific number of "wins" (successes) out of a fixed number of tries.

How it Works

The Binomial Distribution is one of the most useful tools in statistics. It looks at a set number of independent trials where the chance of success is always the same. It helps you predict outcomes when there are only two possibilities, like Pass or Fail.

Key Characteristics

• Fixed Trials (n): You repeat the process a specific number of times.
• Binary Outcome: Each time, the result is either a Success or a Failure.
• Independence: The result of one try does not change the result of the next one.
• Constant Probability (p): The chance of winning stays the same every time.

Real-World Examples

Quality Control: If a factory knows that 1% of their lightbulbs are usually broken, they can calculate the odds that exactly 3 out of 100 bulbs in a box will be defective.

Medical Testing: If a medicine cures 80% of people, doctors can predict how likely it is that 8 out of 10 patients will recover.

The Formula

To find the probability of exactly k successes in n trials:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Quick Summary

The Geometric Distribution calculates how many times you need to try something before you get your first success.

How it Works

This model answers the question: "How long do I have to wait?" It looks at a sequence of independent trials and tells you the probability that your first win will happen on a specific attempt.

When to use it

Use this when you are repeating a process over and over until a specific event happens. The attempts must be independent, and your chance of success must stay the same every time.

Real-World Scenarios

Sales: How many cold calls does a salesperson need to make before making their first sale?

Drilling: How many wells does a company need to drill before hitting oil?

Logins: How many times does a user guess a password before getting it right?

The Formula

The probability that your first success happens on trial number k:

P(X = k) = (1-p)^(k-1) * p

Quick Summary

The Bernoulli Distribution describes a single event that has a Yes or No outcome, like one coin flip or one pass/fail test.

How it Works

This is the simplest building block of probability. It models a single experiment with exactly two possible results: "Success" (1) and "Failure" (0).

Relationships

Think of Bernoulli as the starting point. If you repeat a Bernoulli trial many times, you create a Binomial Distribution. If you keep repeating it until you finally win, that creates a Geometric Distribution.

Real-World Examples

• Coin Flip: A single toss landing on Heads.
• Email Marketing: Whether one specific customer opens your email or ignores it.
• Exams: Whether a student passes or fails a test.

Key Stats

• Mean (Average): Equal to the probability p.
• Variance: Equal to p times (1 minus p).

Quick Summary

The Poisson Distribution predicts how many times an event is likely to happen within a specific period of time or space.

How it Works

Unlike the other calculators here, the Poisson distribution does not have a fixed number of trials. Instead, events simply happen at a certain average rate. It helps you model independent events occurring in a fixed interval.

Key Assumptions

• Events happen independently of each other.
• The average rate (lambda) stays constant over time.
• Two events cannot happen at the exact same instant.

Classic Examples

Call Centers: Estimating the number of support calls an agent will receive in the next hour.

Traffic: Counting the number of cars passing through a toll booth every minute.

Web Servers: Predicting the number of visitors hitting a website every second.

The Parameter (Lambda)

The symbol Lambda represents the average number of events expected in the time frame. A unique feature of Poisson is that the Mean and the Variance are both exactly equal to Lambda.