Question

If a day is a GOOD day, there is 60% chance that the next day will be GOOD and a 40% chance it will be BAD. If a day is a BAD day, there is 70% chance that the next day will be BAD and a 30% chance it will be GOOD. If today is GOOD, on average, how many days do we have to wait until the next GOOD day (e.g. if it happens tomorrow, that would be 1)?

Answer 1

To calculate the average number of days until the next GOOD day, a probability tree is used to determine the probabilities of each outcome. This allows us to find the average by multiplying each outcome by its probability and summing them up. The average is calculated to be approximately 1.8 days.

Step-by-step explanation:

To find the average number of days we have to wait until the next GOOD day, we can use a probability tree. If today is GOOD, there is a 60% chance that tomorrow will also be GOOD. If tomorrow is GOOD, then the average number of days we have to wait until the next GOOD day is 1. If tomorrow is BAD, we have to go through the same process again. If tomorrow is BAD, there is a 70% chance that the day after that will be BAD. If the day after tomorrow is BAD, then the average number of days we have to wait until the next GOOD day is 1 plus the average number of days we have to wait from that point. By continuing this process, we can calculate the average number of days until the next GOOD day.

Using this information, we can build a probability tree:

Based on this probability tree, we can calculate the average number of days until the next GOOD day by multiplying each possible outcome by its corresponding probability and summing them up. In this case, the average number of days until the next GOOD day is:

Average = 1 * 0.6 + (1 + Average) * 0.4 * 0.7 + (1 + Average) * 0.4 * 0.3

Simplifying this equation, we get:

Average = 1 * 0.6 + 0.28 + 0.12A

0.88A = 1.6

A = 1.8

Therefore, the average number of days we have to wait until the next GOOD day is approximately 1.8 days.