There was a new gossip going around the office. The individual who took this information out passed it on with an intensity of one person per hour. Every person who learned of the rumor continues to pass it on at an intensity of three people per hour. How long can you expect the gossip to be known to the majority of the office staff? Assume a population of 20.
I'm a bit lost here. Should I assume intensity of 1 for the first person, and then a joint one for the rest of the people? How do i calculate the time?
Three different scenarios:
If $k$ people know and each passes it on to people who do not know, then the rate of passing on messages is then $3k$ so the expected time for the next new person to know is $\frac{1}{3k}$
If $k$ people know and each passes it on to somebody else who may or may not know, then the rate of passing on messages is then $3k$ and the rate of passing on messages to people who do not know is $3k\frac{20-k}{19}$ so the expected time for the next new person to know is $\frac{19}{60k-3k^2}$
If $k$ people know and each passes it on to somebody else who may or may not know though not to the person who told them, then the rate of passing on messages is then $3k$ and the rate of passing on messages to people who do not know is $3k\frac{20-k}{18}$ (except for $k=1$ where the rate is $3$) so the expected time for the next new person to know is $\frac{6}{20k-k^2}$ (except for $k=1$ where the expected time is $\frac13$)
You want the message to be passed on ten times to new people so the majority, i.e. $11$ people, know. So you have to sum the ten expected times. This gives the respective sums for the three scenarios:
$\sum\limits_{k=1}^{10} \frac{1}{3k} \approx 0.9763$ hours
$\sum\limits_{k=1}^{10} \frac{19}{60k-3k^2} \approx 1.1551$ hours
$\frac13+\sum\limits_{k=2}^{10} \frac{6}{20k-k^2} \approx 1.1119$ hours