The probability of the union of two events A and B is given by the formula P(A B) = P(A) + P(B) - P(A B). We have all the necessary values to compute this.

To find P(A B), we first need P( B). Since P(B) = 0.4, P( B) = 1 - P(B) = 1 - 0.4 = 0.6. Now we can use the union formula for A and B.

We need to find the intersection of event B with the union of A and B. Using the distributive property of set operations, B (A B) = (B A) (B B). Since B B is an empty set (null event), its probability is 0. Therefore, P(B (A B)) = P(A B).

We need to find the intersection of event B with the union of A and B. Using the distributive property of set operations, B (A B) = (B A) (B B). Since B B is an empty set (null event), its probability is 0. Therefore, P(B (A B)) = P(A B).

Finally, we use the formula for conditional probability: P(X Y) = (P(X Y))/(P(Y)). We have already calculated both the numerator and the denominator in the previous steps.

If A and B are two events such that $P(A) = 0.7$ , $P(B) = 0.4$ and $P(A \cap \overline{B}) = 0.5$ , where $\overline{B}$ denotes the complement of B, then $P(B \mid (A \cup \overline{B}))$ is equal to:

Answer: $\frac{0.2}{0.8} = 0.25$

Step-by-step solution

Step 1: Find P(A ∩ B)

We know that the probability of event A can be expressed as the sum of the probability of A intersecting with B and the probability of A intersecting with the complement of B. This is because event A can be partitioned into two disjoint parts: $A \cap B$ and $A \cap \overline{B}$ . Using the given values, we can find $P(A \cap B)$ .

Step 2: Calculate P(A ∪ B)

The probability of the union of two events A and B is given by the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . We have all the necessary values to compute this.

Step 3: Calculate P(A ∪ B̅)

To find $P(A \cup \overline{B})$ , we first need $P(\overline{B})$ . Since $P(B) = 0.4$ , $P(\overline{B}) = 1 - P(B) = 1 - 0.4 = 0.6$ . Now we can use the union formula for $A$ and $\overline{B}$ .

Step 4: Calculate P(B ∩ (A ∪ B̅))

We need to find the intersection of event B with the union of A and $\overline{B}$ . Using the distributive property of set operations, $B \cap (A \cup \overline{B}) = (B \cap A) \cup (B \cap \overline{B})$ . Since $B \cap \overline{B}$ is an empty set (null event), its probability is 0. Therefore, $P(B \cap (A \cup \overline{B})) = P(A \cap B)$ .

Step 5: Calculate P(B | (A ∪ B̅))

Finally, we use the formula for conditional probability: $P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$ . We have already calculated both the numerator and the denominator in the previous steps.

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If A and B are two events such that P(A)=0.7P(A) = 0.7P(A)=0.7, P(B)=0.4P(B) = 0.4P(B)=0.4 and P(A∩B‾)=0.5P(A \cap \overline{B}) = 0.5P(A∩B)=0.5, where B‾\overline{B}B denotes the complement of B, then P(B∣(A∪B‾))P(B \mid (A \cup \overline{B}))P(B∣(A∪B)) is equal to: