4) Dependent Trials and Tree Diagrams
![[img]data:image/png;base64,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[/img]](https://www.geogebra.org/resource/qvawcwte/JiuA9KIy6ZItjzeE/material-qvawcwte.png)
Dependent Events and Tree Diagrams
When the outcome of one event affects the outcome of another event, the events are determined to be dependent events. Events that are not independent are classified as dependent events.
To determine and classify events as independent or dependent, use the relationships P(A|B) = P(A) or P(B|A) = P(B).
Find the Probability of A, P(A), and P(A|B) or the probability of B, P(B), and P(B|A). If the P(A|B) does not equal P(A) or P(B|A) does not equal P(B), the events are dependent.
The probability that two events A and B will happen in sequence uses the Multiplication rule:
P(AnB) = P(A)*P(B|A).
If the element or item is replaced after the first selection the events are independent. If the element or item is not replaced after the first selection the events are dependent.
Tree Diagrams are a visual tool used to find the probability of an outcome by multiplying the probabilities of the branches.
Independent Event: Coin toss tree diagram the outcome of each coin toss is independent of the result of the previous toss.
Dependent Event: Coin toss tree diagram the outcome depends on the event that happened before it.
Questions students should be able to answer:
1) How to determine if two events are independent or dependent?
2) How does replacement and without replacement affect the outcome of a probability of two events?
3) How to create a Tree Diagram?
4) The probability of each iteration of branches has to equal what number?