Statistics Examples

$P (A) = 0.5$ , $P (B) = 0.75$ , $P (B g i v e n A) = 0.75$

Step 1

Two events are independent events when the occurrence of one does not affect the probability of the other. $P (A | B) = P (A)$ and $P (B | A) = P (B)$ .

$P (A | B) = P (A)$

$P (B | A) = P (B)$

Step 2

$P (B | A)$ should equal to $P (B)$ because the occurrence of $A$ should not effect the probability of $B$ for independent events $A$ and $B$ . In this case, $P (B | A) = P (B) = 0.75$ .

$P (B | A) = P (B) = 0.75$

Step 3

Find $P (A | B)$ using Bayes's rule.

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Using the Bayes' rule, $P (A | B) = \frac{P (B | A) P (A)}{P (B)}$ .

$P (A | B) = \frac{P (B | A) P (A)}{P (B)}$

Substitute the given values $P (A) = 0.5$ , $P (B) = 0.75$ , and $P (B | A) = 0.75$ in Bayes' rule.

$P (A | B) = \frac{(0.75) \cdot (0.5)}{0.75}$

Cancel the common factor of $0.75$ .

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Cancel the common factor.

$P (A | B) = \frac{0.75 \cdot 0.5}{0.75}$

Divide $0.5$ by $1$ .

$P (A | B) = 0.5$

Step 4

$P (A | B)$ should equal to $P (A)$ because the occurrence of $B$ should not effect the probability of $A$ for independent events $A$ and $B$ . In this case, $P (A | B) = P (A) = 0.5$ .

$P (A | B) = P (A) = 0.5$

Step 5

$P (A | B) = P (A)$ and $P (B | A) = P (B)$ , which means that $A$ and $B$ are independent events.

A and B are independent events

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