The Hermit's Epidemic

Expected value has very practical applications. For example, it can be used in the study of infectious diseases. The following is an extremely simplified version of such a study. Despite the somewhat unrealistic nature of the problem, it should help you to see how this statistic can be used.

Six (unusually sociable) hermits live on an otherwise deserted island. An infectious disease strikes the island. The disease has a 1-day infectious period and after that the person is immune (cannot get the disease again). Assume one of the hermits gets the disease (maybe from a piece of the Mir space station). He randomly visits one of the other hermits during his infectious period. If the visited hermit has not had the disease, he gets it and is infectious the following day. The visited hermit then visits another hermit. The disease is transmitted until an infectious hermit visits an immune hermit, and the disease dies out. There is one hermit visit per day. Assuming this pattern of behaviour, how many hermits can be expected, on the average, to get the disease?
(from Using Statistics by Travers, Stout, Swift, and Sextro -- p67)


  1. What is the least number of hermits that could get infected?

  2. What is the greatest number of hermits that could get infected?

  3. What sort of model could you use for this problem?

  4. How would you solve this problem analytically?

  5. How would changing the number of hermits on the island affect the expected number of infected hermits?

Please send questions and comments to Jay Hill