Given $P(A)=0.74, P(B)=0.85$ and $P(A \mid B)=0.84$, find the value 0 $P(A$ and $B)$, rounding to the nearest thousandth, if necessary.

Step 1 ：Given \(P(A)=0.74\), \(P(B)=0.85\) and \(P(A \mid B)=0.84\), find the value of \(P(A \text{ and } B)\), rounding to the nearest thousandth, if necessary.

Step 2 ：We need to find the probability of \(P(A \text{ and } B)\), which can be calculated using the conditional probability formula: \(P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}\). We have the values of \(P(A \mid B)\), \(P(A)\), and \(P(B)\), so we can solve for \(P(A \text{ and } B)\).

Step 3 ：\(P(A \mid B) = 0.84\)

Step 4 ：\(P(A) = 0.74\)

Step 5 ：\(P(B) = 0.85\)

Step 6 ：Using the conditional probability formula, we get:

Step 7 ：\(0.84 = \frac{P(A \text{ and } B)}{0.85}\)

Step 8 ：Solving for \(P(A \text{ and } B)\), we get:

Step 9 ：\(P(A \text{ and } B) = 0.84 \times 0.85\)

Step 10 ：\(P(A \text{ and } B) = 0.714\)

Step 11 ：\(\boxed{0.714}\)