Probability problem

Question

a meeting takes place between a diplomat and eleven goverment officials.However,five of the officials are actually spies. If the diplomat gives secret information to six of the attendees at random, what is the probabillity that no secret information was given to the spies?

Answer 1

Marlene's approach was correct (using conditional probabilities), but she multiplied her fractions incorrectly.

(6/11) * (5/10) * (4/9) * (3/8) * (2/7) * (1/6) = 720/332640 = 0.002165

Another way to see this is with the "combinations" operator, often written 'nCr' for the number of ways of choosing r out of n items, where the order doesn't matter. In our case, out of the 11 people present, we're going to choose a group of 6 of them. There are 11C6 = 462 ways of doing this. And of those 462 equally likely, different ways of picking 6 people, exactly 1 of them will consist entirely of non-spies, since we have 6 non-spies and 5 spies. So the probability is 1/462 = 0.002165 again.

Answer 2

If 5 of the 11 officials are spies then the probability that an official who was selected and told the secret was a spy is 5/11 and the probability that he is not a spy is 6/11.

 

When the first official is chosen the probability he is not a spy is 6/11.

When the 2nd official is chosen the probability he int not a spy is 5/10

When the 3rd official is chosen the probability he int not a spy is 4/9

When the 4th official is chosen the probability he int not a spy is 3/8

When the 5th official is chosen the probability he int not a spy is 2/7

When the 6th official is chosen the probability he int not a spy is 1/6

 

So the probability that all six selected officials are not spies is

6/11 • 5/10 • 4/9 • 3/8 • 2/7 • 1/6 = 3840/516780 = 64/8613 = .0074 or 0.74%