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 intersection of two events \( A \) and \( B \) in terms of their individual probabilities and their conditional probabilities. The relevant formula is:

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

Given the information:
- \( P(A) = \frac{7}{20} \)
- \( P(A \cap B) = \frac{49}{400} \)

First, let's solve for \( P(B|A) \):

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

Substitute the known values:

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

To simplify the fraction:

\[ P(B|A) = \frac{\frac{49}{400}}{\frac{7}{20}} = \frac{49}{400} \times \frac{20}{7} = \frac{49 \times 20}{400 \times 7} = \frac{49 \times 20}{400 \times 7} = \frac{49}{140} = \frac{1}{4} \]

So, \( P(B|A) = \frac{1}{4} \).

Next, we use the relationship:

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

Given \( P(A \cap B) \) and \( P(A) \):

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

To solve for \( P(B) \):

\[ P(B) = \frac{\frac{49}{400}}{\frac{7}{20}} = \frac{49}{400} \times \frac{20}{7} = \frac{49 \times 20}{400 \times 7} = \frac{49}{140} = \frac{1}{4} \]

Therefore, the missing probability \( P(B) \) is \( \frac{1}{4} \).

Would you like more details or have any questions?

Here are 5 questions you can ask next:
1. What is the difference between independent and dependent events in probability?
2. How do you calculate conditional probability?
3. Can you explain the concept of mutually exclusive events with examples?
4. How do we apply Bayes' Theorem in probability problems?
5. What are some common mistakes to avoid when calculating probabilities?

**Tip:** Always check that probabilities are between 0 and 1. This helps in verifying the calculations.

Learn how to find the missing probability P(B) using conditional probability formula. Given P(A) = 7/20 and P(A ∩ B) = 49/400, discover step-by-step how to calculate P(B). Suitable for advanced high school students and anyone interested in probability theory.

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