If $P(A)=0.4$ and $P(B/A)=0.5$ then $P(\overline{A}\cup\overline{B})=$

Q: If $P(A)=0.4$ and $P(B/A)=0.5$ then $P(\overline{A}\cup\overline{B})=$

To solve the problem, we need to find $ P(\overline{A} \cup \overline{B}) $. This is the probability that either event $ A $ does not occur or event $ B $ does not occur, or both. Given: - $ P(A) = 0.4 $ - $ P(B|A) = 0.5 $ First, we need to find $ P(B) $ using the conditional probability formula: \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] Rearranging gives: \[ P(A \cap B) = P(B|A) \times P(A) = 0.5 \times 0.4 = 0.2 \] Now, we use the formula for the probability of the union of complements: \[ P(\overline{A} \cup \overline{B}) = 1 - P(A \cap B) \] Substituting the value we found: \[ P(\overline{A} \cup \overline{B}) = 1 - 0.2 = 0.8 \] Thus, the correct answer is Option 3: 0.8. **Explanation for why each option is incorrect:** 1. **Option 1 (0.6):** This option is incorrect because it does not account for the correct calculation of $ P(\overline{A} \cup \overline{B}) $. The calculation shows that the probability is 0.8, not 0.6. 2. **Option 2 (0.7):** This option is incorrect for similar reasons. The calculation of $ P(\overline{A} \cup \overline{B}) $ results in 0.8, not 0.7. 3. **Option 3 (0.8):** This is the correct option, as demonstrated by the calculation above. The probability of the union of the complements is indeed 0.8. 4. **Option 4 (0.9):** This option is incorrect because it overestimates the probability. The correct calculation shows that the probability is 0.8, not 0.9. In conclusion, the correct answer is Option 3, which is supported by the logical and factual calculation of the probability $ P(\overline{A} \cup \overline{B}) $.

Question

Accepted Answer

To solve the problem, we need to find $ P(\overline{A} \cup \overline{B}) $. This is the probability that either event $ A $ does not occur or event $ B $ does not occur, or both.

Given:
- $ P(A) = 0.4 $
- $ P(B|A) = 0.5 $

First, we need to find $ P(B) $ using the conditional probability formula:
$$ P(B|A) = \frac{P(A \cap B)}{P(A)} $$

Rearranging gives:
$$ P(A \cap B) = P(B|A) \times P(A) = 0.5 \times 0.4 = 0.2 $$

Now, we use the formula for the probability of the union of complements:
$$ P(\overline{A} \cup \overline{B}) = 1 - P(A \cap B) $$

Substituting the value we found:
$$ P(\overline{A} \cup \overline{B}) = 1 - 0.2 = 0.8 $$

Thus, the correct answer is Option 3: 0.8.

**Explanation for why each option is incorrect:**

1. **Option 1 (0.6):** This option is incorrect because it does not account for the correct calculation of $ P(\overline{A} \cup \overline{B}) $. The calculation shows that the probability is 0.8, not 0.6.

2. **Option 2 (0.7):** This option is incorrect for similar reasons. The calculation of $ P(\overline{A} \cup \overline{B}) $ results in 0.8, not 0.7.

3. **Option 3 (0.8):** This is the correct option, as demonstrated by the calculation above. The probability of the union of the complements is indeed 0.8.

4. **Option 4 (0.9):** This option is incorrect because it overestimates the probability. The correct calculation shows that the probability is 0.8, not 0.9.

In conclusion, the correct answer is Option 3, which is supported by the logical and factual calculation of the probability $ P(\overline{A} \cup \overline{B}) $.