# Given the probability of an event occuring over a shorter time frame, how to calculate the probability of event occuring within a longer time frame?

I suspect I have bug infestation at home and have placed traps to monitor the situation. According to research, these traps have a 70% success rate to detect an infestation over a seven day period of time (i.e. trapped bug, number of bugs trapped/present in home is irrelevant).

The traps have been placed for 3 weeks now and no bugs have been trapped. Assuming the probability of trap working is uniformly distributed every day, how should I assess the probability of possible infestation inside my home?

If I can view bug being trapped over 7 day period as a discrete event, then I think the probability of my home not having an infestation can be calculated as follow:

Probability of trap not detecting existing infestation per week = 1 - 0.7 = 0.3

Probability of trap not detecting existing infestation 3 weeks in a row = 0.3 * 0.3 * 0.3 = 0.027 = 2.7%

Conclusion: I have a 97.3% chance of not having an infestation

However the part I am not unsure of is that the probability from the research is over a continuous period of time. Can I really simply view trapping a bug over 7 day as a discrete event? I am not really sure what kind of math I need to do that.

However, I would feel a lot more comfortable (not sure if this is justified) is to view the trap catch a bug in a single day as a discrete event. Assuming the distribution of the probability of catch a bug per day is uniformly distributed over the 7 day period, the daily possibility of catch bug can be calculated as this:

Probability of trap not detecting existing infestation per week = 1 - 0.7 = 0.3

Define probability of trap not detecting bug per day as X, then X^7 = 0.3, X=0.841982

Probabilty of trap not detecting existing infestation 3 weeks in a row = 0.841982^21 = 0.027 = 2.7%

Ok so I am back where I started. Is my thinking and method of calculation correct? Or is it bad to calculating probability over continuous time frame as discrete event?

There are two things that I want to point out:

1. Your method of calculation only applies when the underlying distribution is a uniform distribution. In fact, it could be a gamma distribution.

2. If the distribution is uniform:

70% success rate to detect an infestation over a seven day

means that there's 70% chance that the trap can detect an infestation in 7 days, given that you have an infestation. So, it shouldn't be

Conclusion: I have a 97.3% chance of not having an infestation


but it should be there's a 97.3% chance of detecting an infestation, given that you have it.

• Thanks for the feedback. If the probability is in fact a normal curve, how should I extrapolate the 7 day probability into a 21 day probability? Commented Jun 14, 2020 at 4:12
• for #2, like you said, there is a 97.3% chance of detecting an infestation if I have one. So in logic: Event A = I have an infestation. Event B = infestation is detected over 21 days. A -> B. The contrapositive, ~B->~A would be Infestation not detected -> I do not have an infestation. Does that make sense or am I making an error somewhere? Commented Jun 14, 2020 at 4:19
• 2. Not quite. When you said A->B, you're saying the trap would 100% detect an infestation after 21 days if you have it. The correct way to think about it: given that there is a 97.3% chance of detecting an infestation in 21 days if you have one, if the trap doesn't detect anything after 21 days, it means either: a. You don't have an infestation or b. You have an infestation, but your infestation is not detected. P(b) = 2.7% Commented Jun 14, 2020 at 8:21
• 1. I made a mistake taking normal distribution as an example for this case. If you want to find the probability of detecting infestation at a certain day, then use normal. But in your case, you should use Gamma distribution. You'll understand why as soon as you see its shape. In order to find the probability with gamma distribution, you need to know its parameter alpha and beta, then plug alpha, beta, x=21 into f(x) Commented Jun 14, 2020 at 8:39