# Calculating total probability of event in a time period given probabilities of it happening at different time intervals during that time period

I have time intervals and probabilities for an event to occur in these time intervals. I want to calculate the probability of the event happening once or more during the entire period, that is to say with an event $X$, I want to know $P(X \geq 1)$ across all of the time periods specified. The probabilities can be assumed to be independent of each other.

I'll illustrate with example data:

 Time_bin     Probability
0 <= t < 1        70%
1 <= t < 2        30%
2 <= t < 3        10%
3 <= t < 4        50%
4 <= t < 5        20%

Total:         Total:
0 <= t < 5         ?


I have been thinking about using a binomial distribution somehow, but I really don't know how I would go about that. Any pointers would be greatly appreciated.

• Are the events/non-events in the separate time windows independent of each other? – jwimberley Dec 6 '16 at 11:55
• Yes. An event happening in one time window does not affect the chance of it happening in another. Nor for that matter does it not happening in a time window – Cenfus Dec 6 '16 at 12:12
• Your problem is straight-forward probability then. For two independent events, $P(A|B) = P(A) + P(B) - P(A) P(B)$. Just apply this rule to bins 0 and 1, this result 2, that result and 3, etc. – jwimberley Dec 6 '16 at 12:24
• Man do I feel like a plonker, that's so simple. Nevertheless, I didn't see it, so thank you very much for your help. – Cenfus Dec 6 '16 at 12:50

P(1+ events) = 1 - P(0 events) = $$1-\prod_{t=1}^5 (1-p_t)$$