To find $P(A \text{ and } B)$, we can use the formula based on the inclusion-exclusion principle:

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

Step 1: Substitute the given values

We know:

• $P(A) = 0.4$
• $P(B) = 0.26$
• $P(A \text{ or } B) = 0.436$

Substituting these values into the formula:

$0.436 = 0.4 + 0.26 - P(A \text{ and } B)$

Step 2: Solve for $P(A \text{ and } B)$

$0.436 = 0.66 - P(A \text{ and } B)$

Rearrange to isolate $P(A \text{ and } B)$:

$P(A \text{ and } B) = 0.66 - 0.436$

$P(A \text{ and } B) = 0.224$

Answer

The value of $P(A \text{ and } B)$, rounded to the nearest thousandth, is:

$\boxed{0.224}$

Would you like further details on any specific part of this calculation?

Here are some related questions you might find helpful:

1. How would the value change if $P(A \text{ or } B)$ were different?
2. Can $P(A \text{ and } B)$ ever be negative? Why or why not?
3. How does the inclusion-exclusion principle apply to probabilities with more than two events?
4. What does $P(A \text{ and } B) = 0.224$ signify in practical terms?
5. If $A$ and $B$ were independent, how would that affect $P(A \text{ and } B)$?

Tip: Remember, in probability, the sum of individual probabilities often exceeds the probability of either occurring alone due to overlap in outcomes.