I was told the following probability problem:
While doing a math problem today at the contest the probability of Annie, Tom and Karen getting the problem correct first is 1/7, 1/2, and 5/14 respectively. Annie breaks her pencil lead so is out for the question. What is the probability Tom gets done first?
When I first read this problem, I thought, "Easy. With Annie out of the question, the ratio of chance of Tom and Karen getting the problem correct first remains the same, so the probabilities can be normalized, so the probability that Tom gets done first is: (7/6)*(1/2) = 7/12."
But the argument was then made that assuming the same ratio between Tom and Karen is arbitrary and bogus - we cannot know if Tom and Karen will maintain the same probability ratio of getting the problem correct. For example, if Tom was really good at the questions that Annie normally could get, perhaps he'd get all of the answers she'd have gotten: 1/7 + 1/2 = 9/14.
Does this imply that the probability that Tom gets done first is undefined? Can we not make a statement about the probability? Would such a statement at least be an estimate of the "real" probability? And if we knew the "real" probability, couldn't we identify whether Tom got the question correct with 100% accuracy?
I am confused as to what I believe about probability.