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 #67085 – 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 $P(A \ B)$, $c \cup$ where $c \ A$ is the complement of A. State whether A and B are independent. Justify your answer.

Solution

By the addition law of probability,

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

Hence

$$P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.5 + 0.4 - 0.7 = 0.2.$$

By the definition of conditional probability,

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.4} = 0.5.$$

From the statement of the question it is not clear what should be found further. I shall find probabilities of some possible events.

By the complementary rule,

$$P((A \cup B)^c) = 1 - P(A \cup B) = 1 - 0.7 = 0.3;$$ $$P((A \cap B)^c) = 1 - P(A \cap B) = 1 - 0.2 = 0.8.$$

We recall that events $A$ and $B$ are independent if

$$P(A \cap B) = P(A)P(B).$$

In this case $P(A \cap B) = 0.2$ and $P(A)P(B) = 0.5 \cdot 0.4 = 0.2$.

Therefore, $P(A \cap B) = P(A)P(B)$, hence events $A$ and $B$ are independent.

**Answer**: $P(A \mid B) = 0.5$; $P((A \cup B)^c) = 0.3$; $P((A \cap B)^c) = 0.8$; events $A$ and $B$ are independent.

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