Probability Studio
Master the math of chance. Formulas, Permutations, and Dice statistics.
Basic Probability
favorable / total
Complement
Prob of NOT A
Addition Rule
A OR B
Multiplication
A AND B (Indep.)
Conditional
A given B
Permutations
Arrangements (Order Matters)
Combinations
Selections (No Order)
Bayes Theorem
Updating Priors
Don't Trust Your Gut. Trust the Math.
Humans are notoriously bad at estimating probabilities. We see patterns where none exist (Gambler's Fallacy) and ignore base rates (Bayesian errors).
The Gambler's Fallacy
The belief that "due" numbers must appear to balance things out.
If you flip 5 Heads in a row, what is the chance the next is Tails?
The coin has no memory.
The Monty Hall Problem
A game show host asks you to pick a door. He opens an empty door and offers a switch.
Always switch.
Cheat Sheet
Frequently Asked Questions
What is the difference between Permutation and Combination?
Order matters in Permutations (e.g., a lock combination '1-2-3' is different from '3-2-1'). Order does NOT matter in Combinations (e.g., a fruit salad with 'Apple, Banana' is same as 'Banana, Apple'). Use nPr for arrangements and nCr for selections.
What is the Gambler's Fallacy?
The mistaken belief that if an event happens more frequently than normal during a given period, it will happen less frequently in the future (or vice-versa). For example: 'Red came up 5 times in a row, so Black is due.' In reality, the coin/wheel has no memory. The odds are always 50/50.
What is the probability of rolling a 7 with two dice?
It is roughly 16.7% (6/36). There are 6 ways to make a 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). It is the most common roll.
What is the Law of Large Numbers?
It states that as the sample size grows, the average of the results gets closer to the expected value. If you flip a coin 10 times, you might get 70% heads. If you flip it 1,000,000 times, you will get almost exactly 50% heads.
What is Bayes' Theorem?
A formula that describes how to update the probability of a hypothesis based on new evidence. $P(A|B) = P(B|A)P(A) / P(B)$. It's used in medical testing to determine the real chance you're sick given a positive test result.
Is it harder to win the lottery if I pick '1, 2, 3, 4, 5, 6'?
No. Every specific combination has the exact same probability. '1-2-3-4-5-6' is just as likely to be drawn as '5-12-23-34-45-56'. It just looks less random to humans.
What is Independent vs. Dependent probability?
Events are Independent if one doesn't affect the other (flipping a coin twice). Events are Dependent if the first outcome changes the second (drawing an Ace from a deck changes the odds of drawing another Ace because there are fewer cards left).
What is the Monty Hall Problem?
A famous puzzle where you pick one of 3 doors (one prize, two goats). The host opens a loser door and offers a switch. You should always switch. Switching doubles your win probability from 1/3 to 2/3.
Why is 1 - P(failure) useful?
Sometimes it's hard to calculate the probability of 'at least one success'. It's much easier to calculate the probability of 'Zero successes' and subtract it from 1. This is the Complement Rule.
What is Expected Value?
The average outcome of a random variable over the long run. In gambling, if a $1 bet gives you a 10% chance to win $5, the expected value is $(0.10 imes 5) - 1 = -$0.50$. You lose 50 cents on average per game.