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

Probability Trees

• When forecasting, a useful practice is to break the problem down into a sequence of events.

• 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 Auburn” and “Auburn 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

Let’s pick a forecasting challenge question and try to break it down into a probability tree. Make sure that all your branches are mutually exclusive and collectively exhaustive.

Rules for Tree Construction

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

1. If events \(A\) and \(B\) are mutually exclusive, then \(P(A \text{ or } B) = P(A) + P(B)\).
2. 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:

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

Independence

Coin flips are a classic example of independent events:

Practice

Assign probabilities to each branch of your tree. Make sure you adhere to the axioms of probability. Consider whether the events in your tree 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.

\[ P(A \text{ and } B) = P(A) \times P(B|A) \]

Practice

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

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

Wrap Up

Understanding joint probability gives us some insight into the birthday problem we started with:

The joint probability on that top branch starts to get really small as we add more people!

Wrap Up

• Key ideas from today:

  • Breaking down forecasts into a sequence of events (more on this when we discuss Fermi estimation)

  • The axioms of probability

  • Joint probability

• Next time, we dive into conditional probability, and discuss how to update our beliefs in light of evidence.