The Epidemic Problem

A certain flu has a one day infection period and after that a person is immune to the flu. Nine people live on an isolated island. One person catches the flu and randomly contacts one other person while playing golf during her infection period. The second person is infected immediately and contacts another person at random during the next day (i.e. his infection period). This process continues with one contact per day until an infected person contacts an immune person and the flu dies out...

... If the first simulation trial is 6,8,0. i.e. 3 people are infected by this flu before it dies out, then what would the results be if the simulation is to be repeated 30 times?

eawoodall

if you run a actual test your results will hardly ever exactly match the probabilty of a fair result.

but here is the probability of the events occurring. P(day 1)= 1=P(day 2).

let us label the (nine total) people 0 through 8
the first person infected we call patient zero, infects patient one (the second person infected) on day two, then the probability that one (the second person infected) will infect a new person is 7/8 there are only eight other people on the island besides themselves, and so on 6/8,5/8,4/8,3/8,2/8,1/8.

so on the third day there is a 7/8 = 0.875 chance of disease.
new infection on fourth day = 6/8 = 0.75
times previous 7/8 -> 7/8 * 6/8 = 42/64 =0.65625 chance of disease on day 4

new infection on fifth day 5/8 = 0.625
times all previous -> 42/64 * 5/8 = 210/512 =0.41 chance of disease on day 5

new infection on sixth day 4/8 = 0.50
times all previous -> 210/512 * 4/8 = 840/4096 =0.205 chance on day 6

new infection on seventh day 3/8 = 0.375
times all previous -> 840/4096 * 3/8 = 2520/32768 = 0.076904297 = P(day 7)

new infection on eighth day 2/8 = 0.25
times all previous new infections -> 5040/262144 = 0.019226074 = P(day 8).

new infection on ninth day 1/8 = 0.125
times all previous new infections -> 5040/2097152 = 0.002403259 =P(day 9).

it has been a while since you asked this question, but i thought it was a good thing to answer as a general solution to seeing approaches to probability word problems.