Five 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 three NON-IMMUNE hermits (probability=3/5), the disease is passed to a fourth hermit. If he, then, visits one of the two NON-IMMUNE hermits (probability=2/5), the disease is passed to a fifth hermit. If the fifth hermit visits an IMMUNE hermit (probability=4/5), the outbreak ends with five 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 another non-immune hermit (3/5 chance).
Hermit #1 is randomly visited and, because he is not immune, he gets the disease.Hermit #1, then, randomly visits a non-immune hermit (2/5 chance).
Hermit #6 is randomly visited and, because he is not immune, he gets the disease.Hermit #6, then, randomly visits one of the other immune hermits (4/5 chance) and the outbreak ends.


The probability that exactly five hermits get sick = (The probability that the second hermit visits a non-immune hermit) · (The probability that the third hermit visits a non-immune hermit) · (The probability that the fourth hermit visits a non-immune hermit) · (The probability that the fifth hermit visits an immune one) = (4/5) · (3/5) · (2/5) · (4/5) = 96/625 = 0.1536


Return to the analytical solution.

Or continue to the explanation page for 6 infected hermits.