Final answer:
The correct formula for the product rule of probability for three random variables X, Y, and Z is P(a, b, c) = P(a | b c) P(b | c) P(c), which is option D.
Step-by-step explanation:
Among the options provided, the correct answer is D. P( a | b ) P( b | c ) P( c ). This is because the product rule of probability states that the probability of multiple events occurring together is equal to the product of the probability of one event occurring given the previous event has occurred, multiplied by the probability of the previous event occurring. In this case, for three random variables X, Y, and Z, the probability P(a, b, c) denotes the probability that X=a, Y=b, and Z=c, simultaneously.
According to the product rule, this is calculated by the probability of X=a given that Y=b and Z=c, P(a | b c), times the probability of Y=b given that Z=c, P(b | c), times the probability of Z=c, P(c). Mathematically, this can be expressed as:
P(a, b, c) = P(a | b c) × P(b | c) × P(c)
Therefore, the correct answer is the formula that represents the multiplication of these probabilities.