Unit 6 Discrete Probability

Discrete Distribution

A probability distribution where only whole number outcomes are possible (e.g., dice rolls, coin flips).

Conditions for Binomial Distribution

1. Fixed number of trials 2. Only two outcomes (success/failure) 3. Constant probability 4. Independent trials.

Usage of Binomial Distribution

Finding the probability of getting a certain number of successes in repeated trials.

Success in Binomial Distribution

Any outcome you're measuring (e.g., winning, catching a shiny Pokémon, getting sick) – it doesn’t always mean something positive.

Example of Binomial Distribution

Flipping a coin 50 times and calculating the chance of getting exactly 20 heads.

Expected Value of Binomial Distribution

Multiply the number of trials (n) by the probability of success (π): μ = n × π.

Poisson Distribution

A probability model used for counting events in a fixed interval of time or space when events happen at a steady average rate.

When to Use Poisson Distribution

When events happen randomly but at a known average rate, and each event is independent.

Example of Poisson Distribution

Catching fish during a 3-hour boat trip where you usually catch 8 fish on average.

Expected Value of Poisson Distribution

The mean is the average rate (λ or μ), which also equals the variance in a Poisson distribution.

Discrete Variable

A variable with specific, countable values (like 0, 1, 2…).

Continuous Variable

A variable that can take any value within a range, including decimals (e.g., time, weight, temperature).

Rules for Probability Distribution

1. Each probability must be between 0 and 1 2. All probabilities must add up to 1 (100%).

Finding Expected Value for Discrete Variable

Multiply each outcome by its probability, then add all the results together.

Importance of Wording in Binomial Problems

Phrases like “fewer than,” “or more,” and “exactly” affect which values you include when calculating probability.