Final answer:
The probability that John will be the vice president given Erika is the president is 1/2. This is because there are only two options for the vice president (John or Ron) if Erika is the president.
Step-by-step explanation:
To determine the probability that John will be the vice president, we need to consider Erika's constraints. Since Erika will serve only as president and only if either John or Ron are vice presidents, we will look into the scenario where Erika is president.
If Erika is the president, we have two choices for Vice President (John or Ron), and 7 remaining students can be the treasurer. However, if Erika is not the president, any of the remaining 8 students can be president, vice president, or treasurer, without any restriction.
So the probability of Erika being president is the sum of the probabilities of John or Ron being the vice president:
- P(Erika is President and John is VP) = 1/2 (since there are only two choices for the VP position)
- P(Erika is President and Ron is VP) = 1/2
Since both events are mutually exclusive, we can add these probabilities:
P(Erika is President) = P(John is VP) + P(Ron is VP) = 1/2 + 1/2 = 1
However, this is true when Erika is indeed the president. In any other case, John can't be the VP. So, since Erika being president is a requirement and is certain if John or Ron is VP, the probability that John is the vice president given that Erika is the president is simply:
P(John is VP | Erika is President) = 1/2