Final answer:
To find the probability that the two numbers differ by 1, count the favorable outcomes and divide by the total outcomes. To find the probability they do not differ by 1 on exactly eight or nine trials, use the binomial probability formula.
Step-by-step explanation:
The probability that the two numbers differ by 1 can be found by counting the favorable outcomes and dividing by the total number of outcomes. In this case, the favorable outcomes are (1,2) and (3, 2), since the numbers differ by 1. There are 2 favorable outcomes out of a total of 6 possible outcomes, so
the probability is 2/6 = 1/3.
To find the probability that the numbers do not differ by 1 on exactly eight or nine of the 15 occasions, we need to consider the complementary probability. If the numbers do not differ by 1, they must either be the same or differ by 2.
Considering the first spinner, there are three possible outcomes (1,1), (2,2), and (3,3) where the numbers are the same, and two possible outcomes (1,3) and (3,1) where the numbers differ by 2.
Therefore, the probability of not differing by 1 on a single trial is 5/6. The probability of this happening on exactly eight or nine trials out of 15 can be found using the binomial probability formula, which in this case is:
15C8 * (5/6)^8 * (1/6)^7 + 15C9 * (5/6)^9 * (1/6)^6.
If you compute this expression, you will find the probability.