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According to the record of the registrar's office at a state university, 35% of the students are freshman, 25% are sophomore, 16% are junior and the rest are senior. Among the freshmen, sophomores, juniors and seniors, the portion of students who live in the dormitory are, respectively, 80%, 60%, 30% and 20%. If a randomly chosen student does not live in the Dormitory, what is the probability that he is a sophmore ?

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    And your question about the problem is..? – 2017-02-10
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    what is the answer? – 2017-02-10
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    http://meta.math.stackexchange.com/questions/9959/how-to-ask-a-good-question#9960 – 2017-02-10
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    thank you for making fun of my name. – 2017-02-10
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    One little thing: do you really think it's a "real life situation" ? – 2017-02-10
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    This is a question about [conditional probability](https://en.wikipedia.org/wiki/Conditional_probability). Where are you getting stuck? – 2017-02-10
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    To give you an idea of how to proceed, sometimes it's useful to start out with some concrete numbers. Suppose there are $10000$ students at the university (a not unreasonable number). Then $3500$ are first-years, $2500$ are second years, and so on. Then of those $3500$ first-years, $80$ percent, or $2800$, live in the dorm, while the remaining $700$ don't. And so on for the other years. Add up all those who don't live in the dorms. What percentage is represented by second-year non-dorm-livers? – 2017-02-10

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An outline leading to the solution:

According to the definition of conditional probability, $P(So|D^c) = P(So \cap D^c)/P(D^c).$

For the numerator: $P(So \cap D^c) = P(So)P(D^c|So) = .25(1-.6) = 0.1.$

For the denominator: $$P(D^c) = P(F \cap D^c)+P(So \cap D^c)+P(J \cap D^c)+P(Sr \cap D^c).$$ This is an example of the Law of Total Probability.

We have already found one of these four probabilities. Find the other three in a similar way, and add.

Then do the division for the final answer.

This is an example of Bayes' Theorem. (Hence my edit to the title of your question.)

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    Still there? Can you find $P(F \cap D^c)?$ Back in a few hours. – 2017-02-11