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Conditional probability: Tree diagrams
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Tree diagrams are very useful when finding conditional probabilities.
On a tree diagram, the probability on the second branch represents the probability of BBB happening given that AAA has/has not occurred:
Consider an experiment with the following probabilities: P(N)=13P(N)=\dfrac{1}{3}P(N)=31, P(M∣N′)=12P(M|N')=\dfrac{1}{2}P(M∣N′)=21 and P(M′∣N)=23P(M'|N)=\dfrac{2}{3}P(M′∣N)=32. Find P(N∩M)P(N\cap M)P(N∩M).
Draw and complete a tree diagram as follows:
Find P(N∩M)P(N\cap M)P(N∩M) by multiplying the first set of branches:
P(N∩M)=P(N)×P(M∣N)=13×13=19\begin{aligned} P(N \cap M) &= P(N) \times P(M|N) \\\\&= \dfrac 1 3 \times \dfrac 1 3 \\\\&= \dfrac 1 9\end{aligned}P(N∩M)=P(N)×P(M∣N)=31×31=91
So, P(N∩M)=19‾\underline{P(N\cap M)=\dfrac{1}{9}}P(N∩M)=91.
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Overview
The probability of B given A, P(B|A).
The P(A) in P(B|A).
They are used to calculate probabilities and also can be used to calculate conditional probabilities.
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