Final answer:
The probability of a drought lasting exactly 3 intervals is 2%, and the probability of a drought lasting at most 3 intervals is 4.688%.
Step-by-step explanation:
To find the probability that a drought lasts exactly 3 intervals, we need to multiply the probability of a drought occurring (0.20) with the probability of water rationing occurring given a drought (0.10). So the probability is 0.20 * 0.10 = 0.02, or 2%.
To find the probability that a drought lasts at most 3 intervals, we need to find the sum of the probabilities of a drought lasting 1, 2, and 3 intervals. Assuming each interval is independent, the probability of a drought lasting 1 interval is 0.02, the probability of a drought lasting 2 intervals is (0.20 * 0.10) * (0.80 * 0.15) = 0.024, and the probability of a drought lasting 3 intervals is (0.20 * 0.10) * (0.80 * 0.15) * (0.80 * 0.15) = 0.00288. So the probability of a drought lasting at most 3 intervals is 0.02 + 0.024 + 0.00288 = 0.04688, or 4.688%.