To estimate the number of days Barney will be late for college next year, we need to consider the probabilities of him going by bus or walking, as well as the probabilities of him being late in each case.
Let's break down the problem step by step:
1. Probability of going by bus: The probability that Barney will go to college by bus on any given day is 0.3. Since he will go to college on 200 days next year, we can estimate that he will go by bus on 0.3 * 200 = 60 days.
2. Probability of being late when going by bus: The probability that Barney will be late when he goes to college by bus is 0.2. Therefore, on the 60 days he goes by bus, we can estimate that he will be late on 0.2 * 60 = 12 days.
3. Probability of walking: Since Barney either goes by bus or walks, the remaining days (200 - 60 = 140) are the days he walks to college.
4. Probability of being late when walking: The probability that Barney will be late when he walks to college is 0.1. Therefore, on the 140 days he walks, we can estimate that he will be late on 0.1 * 140 = 14 days.
Finally, to estimate the total number of days Barney will be late for college next year, we sum the estimated days for each mode of transportation: 12 days (going by bus) + 14 days (walking) = 26 days.
Therefore, an estimate for the number of days Barney will be late for college next year is 26 days.