Probability with Tree Diagrams: Learn It 3

  • Learn how to construct and interpret a tree diagram to represent sequential events or decisions
  • Calculate conditional probabilities using tree diagrams, considering both dependent and independent events
  • Apply tree diagrams to solve problems involving probability of multiple events, such as probability of compound events and conditional probability

A very natural use of tree diagrams is to help visualize conditional probability for compound events. 

conditional probability

The conditional probability of [latex]B[/latex] given [latex]A[/latex] is denoted as [latex]P(B\text{ given }A) = P(B|A).[/latex]

 

[latex]P(B\text{ given }A) = P(B|A)[/latex] is the probability that event [latex]B[/latex] will occur given that the event [latex]A[/latex] has already occurred.

 

In a tree diagram, conditional probability focuses on a specific branch of this tree, considering that a particular event has already happened.

 

In mathematical terms, if [latex]P(A)[/latex] represents the probability of event [latex]A[/latex], and [latex]P(B)[/latex] represents the probability of event [latex]B[/latex], the conditional probability of [latex]B[/latex] given that event [latex]A[/latex] has occurred is denoted as [latex]P(B|A)[/latex], and it is calculated as:

Utilizing the tree diagram allows us to intuitively grasp conditional probability!

By using the tree diagram, we can visualize the conditional probability [latex]P(B\text{ given } A)[/latex] by analyzing a particular route within the tree. So, when we consider events [latex]A[/latex] and [latex]B[/latex], we can interpret the conditional probability [latex]P(B\text{ given } A)[/latex] by first tracing the path associated with event [latex]A[/latex] and then determining the probability associated with the subsequent path leading to event [latex]B[/latex].

Suppose 10 marbles were placed in a bag with 4 being blue and 6 being red. Suppose two marbles were consecutively drawn without replacement from the bag.

The tree diagram for the given scenario is given below.

This is a tree diagram with branches showing frequencies of each draw. The first branch shows two lines: 4/10 blue and 6/10 red. The second branch has a set of two lines, 3/9 blue and 6/9 red for the blue branch and 4/9 blue and 5/9 red for the red branch.

Let’s calculate the following conditional probability!

(a) If a red was drawn first, what is the probability that the second draw was blue?

(b) If a blue was drawn first, what is the probability that the second draw is red?

(c) If a blue was drawn first, what is the probability that the second draw is also blue?