Unit 1: Probability and Counting
Sample spaces, every way there is to count something, and the two moves — inclusion-exclusion and story proofs — that tame hard counting problems.
13 topics · 49 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
1.1 Sample Spaces and the Naive Definition of Probability
- How it worksHow does listing every outcome in a sample space turn counting into a probability?
- Common mistakeWhy isn't every outcome in a sample space automatically equally likely?
- Worked exampleFind the probability that two rolled dice sum to 8 using the naive definition.
- Going furtherWhat happens to the naive definition of probability once the outcomes stop being equally likely?
Your turn1 of 2
A fair six-sided die is rolled twice and the two results are added. Find the probability that the sum equals 9.1.1 The Multiplication Rule for Counting
- How it worksHow does a sequence of independent choices multiply into one total count?
- Worked exampleFind how many different outfits come from 4 shirts, 3 pairs of pants, and 2 pairs of shoes.
- Going furtherWhat happens to the total outfit count when one shirt can't be paired with one pair of pants?
Your turn1 of 2
A restaurant offers 6 appetizers, 6 entrees, and 8 desserts. How many different 3-course meals can you order, choosing one of each?1.2 Permutations: Counting Orderings
- How it worksHow does the count of orderings collapse from a string of choices down to n factorial?
- Worked exampleFind how many ways 5 sprinters can finish first, second, and third in a race.
- Common mistakeWhy isn't the number of orderings of n objects just n multiplied by 2?
- Ties togetherHow does a permutation turn into a combination once the order stops mattering?
Your turn1 of 2
Find how many ways 15 runners can finish in first, second, and third place, if the order of finish matters.1.2 Combinations and the Binomial Coefficient
- How it worksHow does dividing out the redundant orderings turn a permutation into n choose k?
- Worked exampleFind how many 5-card poker hands can be dealt from a standard 52-card deck.
- Worked exampleFind the number of ways to choose a 3-person committee from a group of 8 people.
- Common mistakeWhy doesn't choosing 3 people for a committee count the same as choosing them for president, VP, and treasurer?
- Going furtherWhat happens to n choose k as k climbs from 0 up past n divided by 2?
Your turn1 of 2
Find the number of ways to choose a 3-person committee from a group of 23 people.1.3 The Four Ways to Sample: Order and Replacement
- How it worksHow do replacement and order split every sampling problem into four counting formulas?
- Common mistakeWhy isn't sampling without replacement just the with-replacement count divided by some fixed number?
- Worked exampleFind how many 4-digit PINs exist if digits can repeat, and how many if they can't.
- Going furtherWhat happens to the four sampling counts as the population being sampled grows very large?
Your turn1 of 2
A lock uses a code of length 4 chosen from 9 possible symbols, and symbols may repeat. How many codes exist?1.3 Bose-Einstein Counting (Stars and Bars)
- How it worksHow do stars and bars turn identical items into gaps you're choosing between?
- Worked exampleFind how many ways to distribute 10 identical candies among 3 children, with some allowed to get none.
- Going furtherWhat happens to the stars-and-bars count when every child must get at least one candy?
Your turn1 of 2
Find how many ways to distribute 18 identical candies among 3 children, with some children allowed to get none.1.4 Story Proofs: Counting the Same Thing Two Ways
- How it worksHow does a story proof pick one set and count it two different ways to prove an identity?
- Worked exampleWalk me through a story proof showing 10 times 9-choose-3 equals 4 times 10-choose-4.
- Common mistakeWhy doesn't proving a counting identity with algebra make it more rigorous than a story proof?
- Ties togetherHow does the story-proof identity connect to Vandermonde's identity for splitting a group in two?
Your turn1 of 2
Using the story-proof identity n times (n-1 choose 2) equals 3 times (n choose 3), find the value of n times (n-1 choose 2) for n = 15.1.4 Vandermonde's Identity
- How it worksHow does conditioning on how many picks come from each half prove Vandermonde's identity?
- Worked exampleFind 10-choose-4 by summing over how many of the 4 picks come from a subgroup of 6 and the rest from 4.
- Going furtherWhat happens to Vandermonde's identity when the group is split into three parts instead of two?
Your turn1 of 2
Using Vandermonde's identity, find 2n choose 2 for n = 18.1.5 The Axioms of Probability
- How it worksHow do nonnegativity, total probability of 1, and additivity build every other probability rule?
- Common mistakeWhy doesn't a valid probability assignment need every outcome to be equally likely?
- Worked exampleShow that assigning probabilities 0.5, 0.3, and 0.1 to three outcomes fails the probability axioms.
- Going furtherWhat happens to the axioms of probability when the sample space has infinitely many outcomes?
Your turn1 of 2
A sample space has three outcomes with probabilities 0.309, 0.295, and x. Using the axiom that total probability is 1, find x.1.5 Properties Derived from the Axioms
- How it worksHow does the axiom of additivity force P(A) plus P(not A) to equal 1?
- Worked exampleFind P(not A) when P(A) equals 0.35, and check it against 1 minus 0.35.
- Common mistakeWhy isn't P(A or B) always just P(A) plus P(B)?
Your turn1 of 2
Find P(not A) when P(A) equals 0.677.1.6 Inclusion-Exclusion for Two and More Events
- How it worksHow does subtracting the overlap turn a double-counted sum into an exact probability?
- Worked exampleFind the probability a card drawn from a standard deck is a heart or a face card.
- Worked exampleFind the probability a random number from 1 to 100 is divisible by 2, 3, or 5.
- Common mistakeWhy doesn't inclusion-exclusion for three events stop after subtracting the three pairwise overlaps?
- Going furtherWhat happens to the inclusion-exclusion formula when there are 10 events instead of 3?
Your turn1 of 2
Events A and B satisfy P(A) = 0.339, P(B) = 0.33, and P(A and B) = 0.143. Find P(A or B).1.7 The Birthday Problem
- How it worksHow does complementing to 'no shared birthday' turn an intractable count into one product?
- Worked exampleFind the probability that at least two people share a birthday in a room of 50 people.
- Common mistakeWhy isn't the birthday problem asking for the chance someone shares your own specific birthday?
- Going furtherWhat happens to the collision probability curve as the room grows from 10 to 100 people?
Your turn1 of 2
A group of 3 people is chosen from a population where birthdays are equally likely among 73 possible days. Find the probability that at least two of them share a birthday.1.7 The Matching Problem and Derangements
- How it worksHow does inclusion-exclusion count the derangements where nobody gets their own tag back?
- Worked exampleFind the probability that none of 5 shuffled name tags return to their original owner.
- Common mistakeWhy doesn't the chance of zero matches keep dropping toward 0 as more people are shuffled?
Your turn1 of 2
15 name tags are shuffled and randomly returned to 15 people. Find the probability that two specific people, Person 1 and Person 2, BOTH get their own tag back.