Practice Problems

Problem 1:

I law of total probability and bayes rule

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Moderate

In the following diagram of sample space, all angles are right angles, and the probability of each event is proportional to its area. Use the law of total probability to compute the value of $P (B)$ : Area Diagram

Solution:

We observe that $A_{1}, A_{2}$ is a partition of $Ω$ . Applying the law of total probability to this partition,

$P (B) = P (A_{1}) P (B | A_{1}) + P (A_{2}) P (B | A_{2})$

Notice that rectangles $A_{1}$ and $A_{2}$ have the same area, so $P (A_{1}) = P (A_{2}) = 1 / 2$ . Rectangle $B \cap A_{1}$ has half the height of rectangle $A_{1}$ and the same width, so $P (B | A_{1}) = P (B \cap A_{1}) / P (A_{1}) = 1 / 2$ . Rectangle $B \cap A_{2}$ has half the height and half the width of rectangle $A_{2}$ , so $P (B | A_{2}) = P (B \cap A_{2}) / P (A_{2}) = 1 / 4$ . Plugging these numbers into the law of total probability above,

$P (B) = (1 / 2) (1 / 2) + (1 / 2) (1 / 4) = 3 / 8$