Probability Trees and Conditional Probability
Probability trees are visual tools that help us organize and calculate probabilities when events happen in sequence. They're especially useful when we need to track multiple possible outcomes across several steps.
What is a Probability Tree?
A probability tree is a diagram that shows all possible outcomes of a sequence of events. It has the following parts:
- Branches: Lines that represent possible outcomes
- Nodes: Points where branches meet or split
- Probabilities: Numbers written on each branch
Each branch in a probability tree has a probability associated with it. The probabilities from any node must add up to 1 (or 100%).
How to Use Probability Trees
To find the probability of a specific sequence of events:
- Follow the path through the tree that represents that sequence
- Multiply the probabilities along that path
This works because when events happen in sequence, we multiply their probabilities to find the combined probability.
What is Conditional Probability?
Conditional probability is the probability that an event will occur given that another event has already occurred. We write it as P(B|A), which means "the probability of event B occurring, given that event A has already occurred."
In a probability tree, conditional probability is represented by the branches that follow after a particular node. The probability on each branch coming from a node tells us the likelihood of that outcome, given that we've already reached that node.
The Conditional Probability Formula
The formula for conditional probability is:
P(B|A) = P(A and B) / P(A)
Where: