Compound Probability

POLS 3220: How to Predict the Future

Warmup

Compound Probability

Last time, we defined probability as a number between 0 and 1 describing the likelihood of an event.
Today, our focus is compound probability, the likelihood of some combination of two or more events.
Compound probability is particularly counterintuitive.
By the end of the lecture, you’ll be equipped with some tools for dealing with these sorts of problems.

Probability Trees

Imagine the future as a series of branching paths.
Each branch represents a different path that the universe could take.

Rules for Tree Construction

Branches originating from a single point must be mutually exclusive and collectively exhaustive.

Mutually Exclusive: if one event happens, the others cannot happen
- e.g. “Georgia beats Texas” and “Texas beats Georgia”.
Collectively Exhaustive: covers every possible outcome
- e.g. “I drink coffee this afternoon” and “I don’t drink coffee this afternoon”.

Rules for Tree Construction

Practice

Think about a sequence of 2-3 choices you expect to make after class is over (e.g. where you’ll eat dinner). Draw a tree illustrating all the possible paths your evening could take.

Rules for Tree Construction

Next, we will assign a probability to each branch. When doing so, remember two rules:

If events \(A\) and \(B\) are mutually exclusive, then \(P(A \text{ or } B) = P(A) + P(B)\).
If events \(A\), \(B\), and \(C\) are mutually exclusive and collectively exhaustive, then \(P(A) + P(B) + P(C) = 1\).

These two rules are called the axioms of probability.

Rules for Tree Construction

The second set of probabilities are called conditional probabilities.

Conditional Probability

We denote the probability of event \(A\) conditional on event \(B\) as:

\[ P(A|B) \]

And the probability of \(A\) conditional on \(B\) not happening is:

\[ P(A|\neg B)\]

Independence

Two events are independent if the outcome of one doesn’t affect the probability of the other. Formally:

\[P(A|B) = P(A|\neg B)\]

Independence

Coin flips are a classic example of independent events:

Practice

Assign probabilities to each branch of your evening tree. Make sure you adhere to the axioms. And consider whether your choices are independent, or if one event might affect the probabilities of subsequent events.

Joint Probability

Now we’re ready to tackle joint probability. What is the probability of event \(A\) and event \(B\) both happening?

Joint Probability

To find the probability of ending up at any node of the tree, multiply the probabilities of all the branches that feed into it.

Practice

Calculate joint probabilities for every node in your evening tree. To check your work, make sure everything satisfies the axioms of probability.

If two events are mutually exclusive, then \(P(A \text{ or } B) = P(A) + P(B)\)
If a set of events are collectively exhaustive, then their probabilities should sum to 100%.

Wrap Up

This gives us some insight into the birthday problem we started with: