Probability Geometry

A meeting takes place between a diplomat and fourteen government officials. However, four of the officials are actually spies. If the diplomat gives secret information to ten of the attendees at random, what is the probability that no secret information was given to the spies? Please show work

Solution

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If you assume that when you give a person some information, he leaves and cannot get anymore information…. that is, they are excluded from the pool once chosen, then you can look at the problem 2 ways.

First, the probability of giving information NOT to a spy is 10/14 for the first pick.

Then for the next pick, the probability the probability of giving information NOT to a spy is 9/13

Taking this through the process you get

10/14 x 9/13 x 8/12 ………….. x 1/5 = 10!/(3632428800) = .000999 which is basically .001 or 1/1000

You can also phrase it like asking what is the probability of choosing the combination of 10 out of 10 divided by the probability of choosing the combination of 10 out of 14 like so…..

P(Choosing No Spies) = 10C10/14C10 = 1/1001