Three Infected Hermits

Each circle represents a hermit. Each red circle represents a hermit who gets the disease. Blue lines are drawn as one hermit visits another. If a hermit turns black, he has been visited twice and the disease ends.

Once the second hermit catches the disease, he visits another hermit. Besides himself, there are five hermits, only one of which is immune. If he DOES NOT visit that one immune hermit (probability=4/5), the disease is passed to a third hermit. That hermit, in turn, visits another hermit. If he visits one of the two immune hermits (probability=2/5), the outbreak ends with three infected hermits.
Hermit #3 randomly gets the disease.	He randomly visits another hermit, who cannot be immune yet.

Hermit #5 is randomly visited and, because he is not immune, he gets the disease.	Hermit #5 randomly visits a non-immune hermit (4/5 chance).

Hermit #2 is randomly visited and, because he is not immune, he gets the disease.	Hermit #2, then, randomly visits an immune hermit (2/5 chance) and, because that hermit is immune, the disease outbreak ends.

The probability that exactly three hermits get sick = (The probability that the second hermit visits a non-immune hermit) · (The probability that the third hermit visits an immune one) = (4/5) · (2/5) = 8/25 = 0.3200

Return to the analytical solution.

Or continue to the explanation page for 4 infected hermits.