Survival bias and probability

Question

Imagine the following situation:

• A new virus is discovered that is believed to have infected 20% of the population.
• Anyone infected with the virus has a chance of 50% of dying in their sleep every day.
• The virus has absolutely no symptoms to let anyone know whether they've been infected or not, before its too late.
• Bob wonders whether he has the virus or not.

Is it possible to calculate how many nights Bob need to survive for him to be 95% sure he does himself not have the virus?

The difficulty I have with this is:

• Survival bias: Noone with the virus will ever wake up and discover they are dead. E.g. Even if the infection rate was 100% you'd have a few random people still waking up after 30 days (1 in 2**30).
• Gamblers fallacy: If you do have the virus, then having survived X days does not make the probability higher than you will survive another day.

Answer 1

What you're describing as survival bias is really the possibility of a false negative, which you have to account for in your calculation. As for the gambler's fallacy, there's no fallacy here if you assume that for infected individuals 50% is always the probability of surviving another day, and that each night's sleep is independent of every other night.

To calculate, use Bayes' rule. Let $V$ be the event that a randomly selected individual has the virus, and let $S_n$ be the event that the person is alive after $n$ nights of sleep. You want to calculate $P(V^c\mid S_n)$ (where $V^c$ is the complement of event $V$), and to find the $n$ such that this probability exceeds $.95$. To apply Bayes, you write $$ P(V^c\mid S_n)={P(S_n\mid V^c)P(V^c)\over P(S_n\mid V^c)P(V^c) + P(S_n\mid V)P(V)} = { 1 \cdot (.8) \over 1 \cdot (.8) + (1/2)^n\cdot (.2)} $$

Answer 2

If you want to know how to solve it, for a question with such a small answer, I recommend using trial and error. Due to the gambler's fallacy, you just need to find out how many infected are left on the planet, and compare your goal probability to the ratio of infected to non-infected. In order for Bob to have a 95% chance to be not infected, approximately 79% of the infected population needs to be killed (leaving 21%).

We need to raise 't' (number of days) until we get a ratio of infected survived $\geq0.21$

$$ Ratio(survived )=.5^t $$

The answer is 3 days, leaving 12.5% of the infected, which would be less than 5% of the remaining population, making Bob sure he is part of the non-infected population.