# Probability and combinatorics

Contents

Basics of probability and combinatorics

## Basic probability

8:18

Intro to probability

We give you an introduction to probability through the example of flipping a quarter and rolling a die.

2:56

Simple probability: yellow marble

In order to find the probability of picking a yellow marble from a bag, we have to first determine the number of possible outcomes and how then many of them meet our constraints.

9:56

Simple probability: non-blue marble

In this example we are figuring out the probability of randomly picking a non-blue marble from a bag. Again, we'll have to think about the possible outcomes first.

Exercise

Simple probability

Practice finding probabilities of events, such as rolling dice, drawing marbles out of a bag, and spinning spinners.

## Venn diagrams and the addition rule

What is the probability of getting a diamond or an ace from a deck of cards? Well I could get a diamond that is not an ace, an ace that is not a diamond, or the ace of diamonds. This tutorial helps us think these types of situations through a bit better (especially with the help of our good friend, the Venn diagram).

10:02

Probability with venn diagrams

Probability of compound events. The Addition Rule.

10:43

Addition rule for probability

Venn diagrams and the addition rule for probability

## Compound probability of independent events using diagrams

What is the probability of making three free throws in a row (LeBron literally asks this in this tutorial).
In this tutorial, we'll explore compound events happening where the probability of one event is not dependent on the outcome of another (compound, independent, events).

5:15

Die rolling probability

We're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes.

2:09

Probability with counting outcomes

The probability of getting exactly 2 heads when flipping three coins. Thinking about this by visualy depicting all of the outcomes.

2:24

Compound events example with tree diagram

Sal figures out the probability of flipping three coins and getting at least two tails.

2:25

Compound events example using diagram

Sal uses a diagram to find the probability of rolling a four-sided die and a six-sided die and not getting a 1.

Exercise

Probabilities of compound events

Practice using sample space diagrams to find probabilities.

## Compound probability of independent events using the multiplication rule

See how we can use the multiplication rule to find the compound probability of independent events.

6:00

Compound probability of independent events

You'll become familiar with the concept of independent events, or that one event in no way affects what happens in the second event. Keep in mind, too, that the sum of the probabilities of all the possible events should equal 1.

8:58

Probability without equally likely events

Up until now, we've looked at probabilities surrounding only equally likely events. What about probabilities when we don't have equally likely events? Say, we have unfair coins?

4:44

Independent events example: test taking

Have you ever taken a test and discovered you have no choice but to guess on a couple of problems? In this example problem, we are considering the probability of two independent events occurring.

2:28

Die rolling probability with independent events

We hope you're not a gambler, but if you had to bet on whether you can roll even numbers three times in a row, you might want to figure this probability first.

9:20

Coin flipping probability

In this video, we 'll explore the probability of getting at least one heads in multiple flips of a fair coin.

7:05

Free throwing probability

Our friend and Cleveland Cavalier, LeBron James, asks Sal how to determine the probability of making 10 free throws in a row. Hint: the answer is surprising!

5:47

Three pointer vs free throwing probability

Our friend and Cleveland Cavalier, LeBron James, asks Sal if there's a high probability of making three free throws in a row or one three-pointer. Before solving the problem, jot down what you think the answer will be!

Exercise

Independent probability

Find probabilities of independent events like flipping a heads and rolling an even number!

## Dependent events

What's the probability of picking two "e" from the bag in scrabble (assuming that I don't replace the tiles). Well, the probability of picking an 'e' on your second try depends on what happened in the first (if you picked an 'e' the first time around, then there is one less 'e' in the bag). This is just one of many, many type of scenarios involving dependent probability.

6:38

Dependent probability introduction

Let's get you started with a great explanation of dependent probability using a scenario involving a casino game.

9:01

Dependent probability: coins

We're thinking about how the probability of an event can be dependent on another event occuring in this example problem.

6:34

Dependent probability example

It's important to practice these probability problems as they get more complex eventually. Take a stab on this one...with our help, of course.

2:39

Independent & dependent probability

This time around we're not going to tell you whether we're working on a dependent or independent probability event problem. You tell us!

Exercise

Dependent probability

Find dependent probabilities like P(A | B) or P(B | A) for a variety of contexts.

7:23

The Monty Hall problem

Here we have a presentation and analysis of the famous thought experiment: the "Monty Hall" problem! This is fun.

5:06

Conditional probability with Bayes Theorem

Conditional probability visualized using trees.

## Permutations

You want to display your Chuck Norris dolls on your desk at school and there is only room for five of them. Unfortunately, you own 50. How many ways can you pick the dolls and arrange them on your desk?

9:01

Factorial and counting seat arrangements

Sal explains a tricky factorial problem about counting seat arrangements.

7:35

Permutation formula

Sal explains the permutation formula and how to use it.

5:39

Possible three letter words

Sal explains how to find all of the possible three letter words when we can use each letter as many times as we want, and when each letter can only be used once.

4:50

Zero factorial or 0!

Sal explains the intuition behind zero factorial.

3:45

Ways to arrange colors

Thinking about how many ways you can pick four colors from a group of 6

2:19

Ways to pick officers

How many ways can we pick officers for our organization?

Exercise

Permutations

Introductory permutation problems.

## Combinations

You are already familiar with calculating permutation ("How many ways could 7 different people sit in 4 different seats?"). But what if you didn't care about which seat they sat in? What if you just cared about which 4 people were in the car? Or put another way, you want to know how many combinations of 4 people can you stick in the car from a pool of 7 candidates. Or how many ways are there to choose 4 things from a pool of 7? Look no further than this tutorial to answer your questions.

6:18

Intro to combinations

Sal introduces the basic idea of combinations.

11:17

Combination formula

Sal explains the combination formula.

7:29

Handshaking combinations

Sal figures out how different combinations of people can shake hands.

7:42

Combination example: 9 card hands

Thinking about how many ways we can construct a hand of 9 cards

Exercise

Combinations

Introductory combination problems like if you have 5 friends and can pick 2 of them to join you on a boat ride, how many different groups of friends could you take with you?

Exercise

Permutations & combinations

Permutations and Combinations with overcounting

## Probability using combinatorics

This tutorial will apply the permutation and combination tools you learned in the last tutorial to problems of probability. You'll finally learn that there may be better "investments" than poring all your money into the Powerball Lottery.

9:10

Probability using combinations

Probability of getting exactly 3 heads in 8 flips of a fair coin.

10:36

Probability & combinations (2 of 2)

Making at least 3 out of 5 free throws.

2:09

Probability with counting outcomes

The probability of getting exactly 2 heads when flipping three coins. Thinking about this by visualy depicting all of the outcomes.

10:00

Getting exactly two heads (combinatorics)

A different way to think about the probability of getting 2 heads in 4 flips

6:51

Exactly three heads in five flips

Probability of exactly 3 heads in 5 flips using combinations

11:40

Generalizing with binomial coefficients (bit advanced)

Conceptual understanding of where the formula for binomial coefficients come from

2:23

Example: Different ways to pick officers

Thinking about the different ways we can pick officers in order to find the probability of one situation in particular.

10:51

Example: Combinatorics and probability

Probability of getting a set of cards

5:10

Example: Lottery probability

What is the probability of winning a 4-number lottery?

Exercise

Probability with permutations and combinations

Probability questions using permutations and combinations of objects

5:30

Mega millions jackpot probability

Probability of winning the Mega Millions jackpot

13:16

Birthday probability problem

The probability that at least 2 people in a room of 30 share the same birthday.