Question

If P(A) = 0⋅50, P(B) = 0⋅ 40 and P(A ∪ B) = 0⋅70,find P(A | B) and
P(A B), c ∪ where c A is the complement of A. State whether A and B are
independent. Justify your answer.

Answer 1

Answer on Question #67021 – Math – Statistics and Probability

Question

If $P(A) = 0.50$, $P(B) = 0.40$ and $P(A \cup B) = 0.70$, find $P(A \mid B)$ and $P(c(A \cup B))$, where $c \mid A$ is the complement of $A$. State whether $A$ and $B$ are independent. Justify your answer.

Solution

By General Addition Rule for probabilities,

$$P(A \cup B) = P(A) + P(B) - P(A \cap B).$$

Therefore,

$$P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.50 + 0.40 - 0.70 = 0.20.$$

By the definition of conditional probability,

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.20}{0.40} = 0.50.$$

By the complementary rule,

$$P(c(A \cup B)) = 1 - P(A \cup B) = 1 - 0.70 = 0.30.$$

For the given events $A$ and $B$, $P(A \cap B) = 0.20$ and $P(A)P(B) = 0.50 \cdot 0.40 = 0.20$. Hence, $P(A \cap B) = P(A)P(B)$ and events $A$ and $B$ are independent.

**Answer**: $P(A \mid B) = 0.50$, $P(c(A \cup B)) = 0.30$, events $A$ and $B$ are independent.

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