Probability Models and Axioms

MIT 6.041 Probabilistic Systems Analysis and Applied Probability, lecture 1, explained in plain English

Want this lecture to stick? In LessonBound you get checked practice questions for every part, can ask about anything you don't get, and can listen with your screen off. Free, no ads.

Learn this lecture free

Watch or read the original lecture: MIT OpenCourseWare.

Part 1 · 0:00–10:00

This opening part of the MIT probability class explains how the course runs, how to study for it, and why probability matters for almost any field.

Key ideas

The course is built around problem solving. Reading and lectures help, but you learn most by solving problems yourself in small tutorial groups with a TA. 1:57
The hard part of probability is not doing the arithmetic. It is choosing the right formula and understanding what it means, which takes judgment and intuition. 6:21
Life is full of uncertainty, and probability gives you a systematic way to model and think about it. Engineers face random noise, managers face random demand, and the same basic tools apply to all of these. 7:05
You need to read the text, not just skim the formulas. Early material is easier, but it gets harder after the first quiz. 9:52

Words to know

mnemonic: A memory aid. The lecturer suggests the slides work as a memory aid that summarizes what happens in class.
tutorial: A small group session with a TA where you work on problems yourself, with help from the TA and classmates.
noise: Random, unwanted signals or interference, such as static on a phone line. In the lecture, engineers must model and deal with it.
plug and chug: Following a formula step by step by putting in numbers, without really understanding why it works. The lecturer says this course is not like that.

Example

Think of a school swap where 10% of students end up unhappy with their class. The head teacher then fixes the schedules again, and 10% of those unhappy students stay unhappy. Only 1% of all students are still unhappy at the end, which is the kind of uncertainty math helps you work out.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part explains how to build a sample space, the full list of things that can happen in a random experiment, and how to describe it clearly.

Key ideas

A sample space is the set of all possible outcomes of an experiment. An experiment is just something that happens, like flipping a coin or rolling a die. 11:04
A good sample space is mutually exclusive and collectively exhaustive. That means every possible result is listed, and exactly one listed outcome happens each time. 12:33
You get to choose how much detail to include, such as whether the weather matters. Choosing what to keep and what to ignore takes judgment. 13:19
When an experiment has several stages, like rolling a die twice, the order of results matters, so a 2 followed by a 3 is a different outcome from a 3 followed by a 2. That gives 16 outcomes in the two-roll example. 16:12
A tree diagram shows each stage in order. Each path from start to end is one overall outcome of the whole experiment. 17:39

Words to know

sample space: The full list (set) of every result that could possibly happen in an experiment.
mutually exclusive: Two results cannot both happen at once. If one happens, the other cannot.
collectively exhaustive: The list covers everything. No possible result is left out, so one of the listed results must happen.
outcome: The final result of the whole experiment, after all of its stages are put together.

Example

Think of a two-stage experiment: pick a card, then flip a coin. The sample space must list every combination, like 'ace then heads' and 'ace then tails', and you only need to count the pieces that matter for your question. If you only care about the card, you can leave the coin out, but then your list is a different sample space.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part explains why we give probabilities to groups of outcomes called events, and the basic ground rules (axioms) any sound probability model must follow.

Key ideas

A single exact point in a continuous target has zero probability, so assigning probabilities to single outcomes tells us very little. We assign probabilities to subsets of the sample space instead. 22:11
A subset of the sample space is called an event. An event occurs if the outcome lands inside it. Probabilities are numbers between 0 and 1 that show how likely an event is. 23:42
The axioms are the ground rules any probability model must follow. The second one says the probability of the whole sample space is 1, because some outcome is certain to happen. 25:06
The third axiom says that if two events share no outcomes, the chance that either one happens is the sum of their separate chances. 27:23
Probability behaves like cream cheese or mass: spread over the sample space, separate pieces add up. The lesson starts deriving the rule that probabilities are at most 1, but the derivation is not finished in this part. 28:05

Words to know

event: A group of possible outcomes, or a subset of the sample space. It happens if the actual outcome is inside that group.
sample space: The full list (or set) of all possible outcomes of an experiment.
axioms: Basic ground rules that you accept as true and build everything else on. For probability, they are the rules any sound model must obey.
union: The combined set of outcomes in either event, or in both. In everyday words, it is the event that A happens or B happens.

Example

Think of a pound of cream cheese spread over a tray. The whole tray holds one pound, which matches the rule that the whole sample space has probability 1. If two separate patches of the tray don't overlap, the cheese on both patches together is just the cheese on one plus the cheese on the other.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part shows how three simple rules of probability lead to useful results, such as why probabilities never exceed one, and then applies them to a rolling-dice example.

Key ideas

The complement of an event A is everything in the sample space that is not in A. Together, A and its complement cover the whole sample space. 30:00
Because A and its complement do not overlap, their probabilities add up to 1. Since probabilities are never negative, every probability must be 1 or less. 30:44
To find the probability of a union of three sets, combine two sets first, then add the third. This means the addition rule for two sets works for any number of disjoint sets. 33:39
Some very strange subsets cannot be given probabilities without breaking the rules. The lesson says you can safely ignore this for the course. 36:35
With a die model where every outcome has probability 1/16, the probability of any event is just the number of outcomes in it times 1/16. 39:32

Words to know

complement: Everything in the full set that is not in the chosen set. If A is an event, its complement is all the outcomes where A does not happen.
disjoint: Two sets with no elements in common. They cannot both happen at the same outcome.
event: A set of possible outcomes that we care about, such as rolls where the first die shows a one.
sample space: The full list of all possible outcomes of an experiment, such as all 16 pairs from rolling a four-sided die twice.

Example

Imagine a bag with 16 marbles, each equally likely to be picked. If 4 of the marbles are red, the chance of picking a red one is 4 out of 16. Each marble counts the same, so you just count the red marbles and divide by 16.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–51:10

This part shows how to find probabilities by drawing a picture of all possible outcomes, counting or measuring the ones you care about, and adding a rule (countable additivity) so that infinite cases also work.

Key ideas

Start by listing every possible outcome in the sample space, which is the set of all things that could happen. Then find the outcomes that make up the event you were asked about. 41:47
When every outcome is equally likely, the probability of an event is just the number of outcomes in the event divided by the total number of outcomes. This is called the discrete uniform law, and it turns many problems into counting. 42:33
For continuous problems like the dart, the probability of a region is its area. A single point has zero area, so its probability is zero, while a triangle's probability comes from its area. 44:21
Adding the probabilities of a list of separate outcomes works for a finite list, but an infinite list needs an extra rule called the countable additivity axiom. That rule is what makes the coin-flip answer of 1/3 valid. 49:26

Words to know

sample space: The full list of all the results that could possibly happen in an experiment.
probability law: A rule that assigns a chance to each possible outcome or group of outcomes in the sample space.
discrete uniform law: The special case where every outcome has the same chance, so you find the answer by counting outcomes.
countable additivity axiom: A rule that says you can add the chances of an endless list of separate, non-overlapping events to get the chance that any of them happens.

Example

Imagine a bag with 16 marbles, and each marble is just as likely to be pulled out as any other. To find the chance of pulling a red one, you count the red marbles and divide by 16, the total number of marbles. This is the same counting idea the lesson uses for dice rolls.

3 checked practice questions for this part are in the app.

Lecture 2: Conditioning and Bayes' Rule →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

LessonBound · Free lecture notes · Privacy · Terms · New York