Question

given the following probabilities determine whether each pair of events is independent or dependent and justify your answer.

• \\(p(a \cap b) = 0.15\\), \\(p(a) = 0.3\\), and \\(p(b) = 0.5\\).
• \\(p(c \cap d) = 0.36\\), \\(p(c) = 0.6\\), and \\(p(d) = 0.4\\).

use the drop-down arrows to complete the sentences.

events \\(a\\) and \\(b\\) are drop-down. this is because \\(p(a|b) =\\) drop-down and \\(p(b|a) =\\) drop-down.

events \\(c\\) and \\(d\\) are drop-down. this is because \\(p(c|d) =\\) drop-down and \\(p(d|c) =\\) drop-down.

Explanation:

Calculate conditional probabilities for A and B

Using the Conditional Probability knowledge point

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.15}{0.5} = 0.3 $$

$$ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.15}{0.3} = 0.5 $$

Determine independence of A and B

Since \(P(A|B) = P(A) = 0.3\) and \(P(B|A) = P(B) = 0.5\), the occurrence of one event does not affect the probability of the other. Therefore, events \(A\) and \(B\) are independent.

Calculate conditional probabilities for C and D

Using the Conditional Probability knowledge point

$$ P(C|D) = \frac{P(C \cap D)}{P(D)} = \frac{0.36}{0.4} = 0.9 $$

$$ P(D|C) = \frac{P(C \cap D)}{P(C)} = \frac{0.36}{0.6} = 0.6 $$

Determine independence of C and D

Since \(P(C|D) = 0.9
eq P(C)\) (where \(P(C) = 0.6\)) and \(P(D|C) = 0.6
eq P(D)\) (where \(P(D) = 0.4\)), the occurrence of one event affects the probability of the other. Therefore, events \(C\) and \(D\) are dependent.

Answer:

Question 1

Events \(A\) and \(B\) are independent. This is because \(P(A|B) =\) \(0.3\) and \(P(B|A) =\) \(0.5\).

Question 2

Events \(C\) and \(D\) are dependent. This is because \(P(C|D) =\) \(0.9\) and \(P(D|C) =\) \(0.6\).