To solve this problem, we will use the formula for the union of two probabilities:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

From the given information:

• $P(A) = \frac{7}{20}$,
• $P(A \text{ or } B) = \frac{231}{400}$,
• We need to find $P(B)$.

Assume $P(A \text{ and } B) = x$. The formula then becomes:

$\frac{231}{400} = \frac{7}{20} + P(B) - x$

Step 1: Convert $P(A)$ to a denominator of 400

$P(A) = \frac{7}{20} = \frac{140}{400}.$

The equation now becomes: $\frac{231}{400} = \frac{140}{400} + P(B) - x.$

Step 2: Rearrange the equation

Rearranging for $P(B)$: $P(B) = \frac{231}{400} - \frac{140}{400} + x.$

Simplify: $P(B) = \frac{91}{400} + x.$

Step 3: Solve for $P(A \text{ and } B)$ if more information is given

To fully solve for $P(B)$, we would need the value of $P(A \text{ and } B)$ (denoted as $x$). If this is independent or additional information is provided, we can substitute it to find $P(B)$.

Would you like me to assume independence ($P(A \text{ and } B) = P(A) \cdot P(B)$) or do you have additional information about $P(A \text{ and } B)$?

Would you like further clarification or a deeper explanation of any steps? Here are 5 related questions to explore:

1. What is the impact of independence on this type of problem?
2. How is the formula for union probability derived?
3. What are the possible ranges for $P(B)$ in this case?
4. How does knowing $P(A \text{ and } B)$ help us finalize $P(B)$?
5. Can you show a real-life example of using this formula in practice?

Tip: When solving probability problems, always check whether the events are independent or dependent, as this greatly affects the calculation.