Question

assignment #5 - mutually exclusive
-> not occurring at the same time
1 - 2, determine whether the events are mutually exclusive.

1. $p(a) = \frac{1}{2}$; $p(b) = \frac{2}{5}$; $p(a \text{ and } b) = 0$
2. $p(a) = \frac{11}{20}$; $p(b) = \frac{3}{10}$; $p(a \cup b) = \frac{9}{20}$

Explanation:

Step1: Recall mutual exclusivity rule

Two events are mutually exclusive if $P(A \cap B) = 0$, or using the addition rule: $P(A \cup B) = P(A) + P(B)$.
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For Problem 1:

Step1: Check $P(A \cap B)$ value

Given $P(A \cap B) = 0$

Step2: Apply mutual exclusivity definition

If $P(A \cap B) = 0$, events are mutually exclusive.
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For Problem 2:

Step1: Calculate $P(A) + P(B)$

$P(A) + P(B) = \frac{11}{20} + \frac{3}{10} = \frac{11}{20} + \frac{6}{20} = \frac{17}{20}$

Step2: Compare to $P(A \cup B)$

Given $P(A \cup B) = \frac{9}{20}$, and $\frac{17}{20}
eq \frac{9}{20}$. Since $P(A \cup B)
eq P(A) + P(B)$, $P(A \cap B)
eq 0$.

Answer:

1. The events are mutually exclusive.
2. The events are not mutually exclusive.