Final answer:
Posterior probabilities are conditional probabilities, which depend on another event having occurred. Conditional probabilities reduce the sample space and are calculated with the formula P(A|B) = P(A and B) / P(B), where P(B) is greater than zero. This contrasts with simpler marginal frequencies and more complex joint probabilities.
Step-by-step explanation:
Posterior probabilities are conditional probabilities. The correct answer is option b. Conditional probabilities occur when the probability of an event or outcome is dependent on another event or outcome having occurred. For example, in probability trees, the conditional probability of drawing a red (R) marble on a second draw, given that a blue (B) marble was drawn in the first, is represented as P(R on 2nd | B on 1st). This probability is based on a restricted sample space where the outcome of the first draw has already been determined.
For instance, if the probability of drawing a blue marble on the first draw is not relevant to the subsequent result, and you know that P(R on 2nd | B on 1st) is 10/30, this shows the reduced sample space after drawing a blue marble. The formula for conditional probability is P(A|B) = P(A and B) / P(B), where P(B) is greater than zero. With this formula, if you were to know P(A and B) and P(B), you could calculate the conditional probability.
Understanding the difference between joint frequencies, marginal frequencies, and conditional probabilities is crucial in handling probabilities. A marginal frequency is a simple probability that tells us the proportion of the entire population fitting a certain category, without conditioning on another event. Joint probabilities involve the probability of two or more events happening at the same time, while conditional probabilities give the probability of one event under the condition that another event has occurred.