Although one revision of prior probabilities can provide useful posterior probability estimates, additional information can be gained from performing the experiment a second time. If it is financially worthwhile, a decision maker may even decide to make several more revisions.

Returning to the previous example, we now attempt to obtain further information about the posterior probabilities as to whether the die just rolled is fair or loaded. To do so, let us toss the die a second time. Again, we roll a 3. What are the further revised probabilities?

To answer this question, we proceed as before, with only one exception. The probabilities $P\left(\text{fair}\right)=0.50$ and $P\left(\text{loaded}\right)=0.50$ remain the same, but now we must compute $P\left(3,3|\text{fair}\right)=$ $\left(0.166\right)\left(0.166\right)=0.027$ and $P\left(3,3|\text{loaded}\right)=\left(0.6\right)\left(0.6\right)=\mathrm{0.36.}$ With these joint probabilities of two 3s on successive rolls, given the two types of dice, we may revise the probabilities:

Thus, the probability of rolling two 3s, a marginal probability, is $0.013+0.18=0.193,$ the sum of the two joint probabilities:

What has this second roll accomplished? Before we rolled the die the first time, we knew only that there was a 0.50 probability that it was either fair or loaded. When the first die was rolled in the previous example, we were able to revise these probabilities:

Now, after the second roll in this example, our refined revisions tell us that

This type of information can be extremely valuable in business decision making.