Two dice are thrown and the outcome is 9. What is the probability distribution of the points on the first die conditional on this given total? Why is this result not incompatible with the obvious fact that the two dice are independent?

When tossing two dice, the number of points on the first die can vary and the way in which these numbers turn up is described by a probability distribution. This distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes. In the given exercise, after knowing that the sum of dice rolls is 9, we analyze the conditional probability distribution of the points on the first die.

This probability distribution, which we calculate only after one condition is met (the sum being 9), reflects the likelihood of each outcome from 1 to 6 on the first die, but only for those outcomes that contribute to the total sum of 9. The conditional distribution is not uniform – for example, there are no combinations where the first die results in 1 or 2 that will yield a sum of 9, so their probabilities are zero. Meanwhile, other numbers like 3, 4, 5, and 6 each have a probability of 1/4, as they each are part of one favorable outcome from the four possible combinations (3,6), (4,5), (5,4), and (6,3).

The exercise dealing with dice rolls is an example of a stochastic process. A stochastic process is a collection of random variables representing a process that evolves over time in a probabilistic manner. Each dice roll can be thought of as one time step in this process, and the outcome is a random variable with its own probability distribution.

More specifically, when dealing with two dice being thrown, we are examining a discrete-time stochastic process with a finite state space. Each dice roll result is a sample point in this space. The probability distribution we discussed earlier is central in defining the characteristics of a stochastic process. Here, the individual outcomes of rolling the dice are independently and identically distributed random variables when considered separately. However, when we impose a condition such as the sum being 9, we are essentially filtering the process and examining it from a particular scenario or outcome.

In probability, independence means that the outcome of one event does not influence the outcome of another event. For dice, this means that what you roll on the first die has no effect on what you roll on the second die and vice versa. In our exercise, before any dice are rolled, all outcomes for both dice are equally likely, demonstrating their independence.

However, once the sum of the dice is known to be 9, we have additional information that alters the sample space to only those outcomes that result in this sum. At this point, we calculate the conditional probability for the first die, but this does not mean the dice have suddenly become dependent. They were rolled independently; it's our post-roll analysis that is conditional. The concept of independence remains intact because the outcome of the sum did not influence the process of rolling the dice, it simply allowed us to narrow down the outcomes after the fact.