An experimental drug was given to a sample of 112 hospital patients with an unknown sickness. Of the total, 76 patients reacted favorably, 15 reacted unfavorably, and 21 were unaffected by the drug. Assume that this sample is representative of the entire population. If this drug is given to Mr. and Mrs. Rivera and their son Carlos, what is the probability that all three react favorably?
(Assume independence.)

The probability is $\square$.
(Round to the nearest thousandth.)

Step 1 ：An experimental drug was given to a sample of 112 hospital patients with an unknown sickness. Of the total, 76 patients reacted favorably, 15 reacted unfavorably, and 21 were unaffected by the drug. Assume that this sample is representative of the entire population.

Step 2 ：If this drug is given to Mr. and Mrs. Rivera and their son Carlos, we are asked to find the probability that all three react favorably. We are assuming independence.

Step 3 ：First, we calculate the probability of a single individual reacting favorably. This is given by the ratio of the number of patients who reacted favorably to the total number of patients. In this case, it is \(\frac{76}{112} = 0.6785714285714286\).

Step 4 ：Next, we calculate the probability of all three individuals reacting favorably. Since we are assuming independence, this is simply the cube of the probability of a single individual reacting favorably. So, \(0.6785714285714286^3 = 0.312\).

Step 5 ：Finally, we round this probability to the nearest thousandth to get \(\boxed{0.312}\).