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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.

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up vote 1 down vote accepted

Of course, math questions like this require you to set the framework of the model. In this case, if we make the assumption that their problem solving abilities are independent of each other, we can proceed like you initially suggested. If you want to add more confounding variables, then the solution becomes more involved.

Your concerns will extend to questions like "If 3 workmen can build 4 benches in 5 days, how long will it take 5 workmen to build 6 benches?" In order to answer the question, we have to make the assumption that they have the same rate of work. It is entirely possible that "5 workmen will build 6 benches in 7 days", because they are lazy workmen.

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Thanks for the response, Calvin. What you describe makes a lot of sense. In particular, though, are making assumptions about, for example, independence "correct" in the context of calculating probability? When is it appropriate to add confounding variables? Are all probabilities subject to the assumptions an individual decides to adopt? – Kirby Mar 16 '13 at 0:33
@Kirby Given that you do not have any further information, most people will say that the assumption of independence is perfectly reasonable. That would be my route, if I had to give a single value answer. Whereas, if this was more of a project, then I would want to consider the other factors that can affect the problem. You started with given bounds on the exact value of the probability, which is great. – Calvin Lin Mar 16 '13 at 0:40
Thanks, Calvin. I think that makes sense. – Kirby Mar 17 '13 at 20:49

Good answer @Calvin Lin

It is worth noting that this failure to understand when real world events are truly independent (rarely) and when they are correlated or even causal can have serious consequences.

There is a case in the UK where a woman spent many years in jail convicted on the testimony of an expert witness who stated (figures are not correct) that the incidence of sudden infant death syndrome (SIDS) were say 1 in 10,000 and that therefore the chance that the woman's 3 children died of SIDS is therefore 1 in 1,000,000,000,000. Aside from the fact that showing they did not die of SIDS does not prove that the mother killed them, it's just plain wrong. The causes of SIDS are unknown but it is conceivable, even likely, that the causes are genetic and/or environmental - siblings share both of these factors so the events are likely to be correlated.

On the flip side, the gambling industry (at least for casino games) largely exists because people believe that largely independent events, like rolls of dice, are in some way correlated. People's runs of bad luck CAN end, but only if they stop gambling!

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