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Probability Distribution

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Probability Distribution

Study Guide

Key Definition

A probability distribution is a function or a table that maps each outcome in the sample space to its probability. For example, when tossing a coin twice, the number of tails X = 0, 1, 2 has probabilities P(X=0)=1/4, P(X=1)=1/2, and P(X=2)=1/4.

Important Notes

Probability distributions map outcomes to their probabilities.
Discrete distributions have countable outcomes.
Each outcome in a sample space must have a probability.
The sum of probabilities for all outcomes equals 1.
Binomial distributions are a type of discrete distribution.

Mathematical Notation

$P(A)$ probability of event A

$P(X = k)$ probability that random variable X equals k

$Σ_k P(X = k) = 1$ sum of discrete probabilities

$nCr = \frac{n!}{k!(n-k)!}$ combinations

$!$ factorial

$f(x)$ probability density function for continuous variables

Use these probability-specific symbols.

Why It Works

Probability distributions help in predicting the likelihood of various outcomes in an experiment. This framework allows calculation of expected values, variances, and underpins results such as the law of large numbers.

Remember

Each outcome in a probability distribution has a weight, and the sum of all these weights (probabilities) must be 1.

Quick Reference

Discrete Probability:$P(X=x) = \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k}$

Understanding Probability Distribution

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Video explanation of this concept

Beginner

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Beginner Explanation

A probability distribution assigns a probability to each outcome of a random experiment. For a fair six-sided die, each face 1–6 has probability $1/6$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the probability of getting heads at least once when tossing a coin twice? $X = 0, 1, 2$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are tossing a fair die. What is the probability of rolling a number greater than 4? $P(X \gt 4)$

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a bag with 3 red, 2 blue, and 5 green marbles. What is the probability of picking a red marble? $P(\text{Red})$

Challenge Quiz

Single Choice Quiz

Advanced

A box contains 4 red, 3 blue, and 3 yellow balls. What is the probability of picking two red balls in a row without replacement? $P(\text{Two Reds})$

Recap

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