Hi Diane! At face value (so to speak), you might think the probability is 100%. If the judge is correct 9 times out of 10 (which is 90%), then in 10 more decisions he's likely to make 1 mistake, right? However, the possibility also exists that he could
make 0 mistakes, so you're actually looking for the odds that he errs 0 or 1 times out of 10.
As you may know, the binomial distribution yields the probability of the number of successes in a sequence of independent yes/no observations. You're looking for failures (incorrect verdicts), but that's just the number of court decisions (10) minus the
number of successes. The formula for the probability of successes is:
P = n! pk(1-p)(n-k)
where n = the total number of observations, k = number of successes, and p = probability. In your example, n = 10, k = 9 or 10, and p = 90%. In other words, you're looking for the probability of 9 or 10 correct decisions ("fewer than 2 incorrect verdicts")
in the next 10 decisions, when the chance of a correct decision is 90%.
There are a couple of ways that you can solve for the probability. One approach is to calculate the probability of 9 successes, then calculate the probability of 10 successes, and then add those two probabilities together to see if the sum matches 0.736.
You'll need to plug-in your values into the above formula, using (for 9 successes) 10, 9, and 0.9 for the variables n, k, and p respectively, and then using (for 10 successes) 10, 10, and 0.9. Remember that ! means factorial, so 10! will be 10*9*8*7*6*5*4*3*2*1.
An alternative is to use Excel to build a table that looks like this:
where you fill-in the values for P using the formula shown above. For example, if 0 (in the above table) is in cell A2, then an appropriate formula would be
=(FACT(10) / (FACT(A2)*FACT(10-A2))) * 0.9^A2 * (1-0.9)^(10-A2)
since n = 10, p = 0.9, and k is in the table (starting with cell A2). Remember to sum the probabilities for 9 successes and for 10 successes, which you'll compare to the given answer of 0.736.
Hope that helps!