Find the missing probability.

P(A)=7/20,P(A∩B)=49/400,P(B)=?

To find the missing probability \( P(B) \), we can use the formula for the probability of the intersection of two events:

\[ P(A \cap B) = P(A) \cdot P(B | A) \]

Given that \( P(A) = \frac{7}{20} \) and \( P(A \cap B) = \frac{49}{400} \), we can rearrange the formula to solve for \( P(B | A) \):

\[ P(B | A) = \frac{P(A \cap B)}{P(A)} \]

Substituting the given values:

\[ P(B | A) = \frac{\frac{49}{400}}{\frac{7}{20}} \]

\[ P(B | A) = \frac{49}{400} \times \frac{20}{7} \]

\[ P(B | A) = \frac{49 \times 20}{400 \times 7} \]

\[ P(B | A) = \frac{49 \times 20}{2800} \]

\[ P(B | A) = \frac{980}{2800} \]

\[ P(B | A) = \frac{49}{140} \]

\[ P(B | A) = \frac{7}{20} \]

Since \( P(B | A) = \frac{7}{20} \), we can find \( P(B) \) using the formula for conditional probability:

\[ P(B | A) = \frac{P(B)}{P(A)} \]

Rearranging to solve for \( P(B) \):

\[ P(B) = P(B | A) \cdot P(A) \]

Substituting the known values:

\[ P(B) = \frac{7}{20} \times \frac{7}{20} \]

\[ P(B) = \frac{49}{400} \]

Thus, \( P(B) = \frac{49}{400} \).

Would you like further details or have any questions about this solution?

Here are 5 more questions you might want to ask:
1. How do you find the probability of the union of two events?
2. Can you explain Bayes' Theorem?
3. What is the difference between independent and dependent events?
4. How do you calculate the expected value of a random variable?
5. Can you demonstrate the use of the Law of Total Probability?

**Tip:** When dealing with probabilities, always ensure the events are properly defined and check whether they are independent or dependent, as this significantly affects the calculations.

Learn how to find the missing probability P(B) given P(A) = 7/20 and P(A ∩ B) = 49/400. This problem involves calculating conditional probability and uses the formula for the intersection of events. Suitable for advanced high school and college students.

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