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I want to calculate the probability of at least one event happening in a series of multiple events. For example, let's say the probability of each event happening are:

  • Event 1: 2/21
  • Event 2: 1/10
  • Event 3: 7/15
  • Event 4: 9/16
  • Event 5: 3/10

What is the probability that at least one of these events will happen?

EDIT: Assume all events are independent.

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(1- the prob. that none happen). If your events are independent, you'd subtract each probability from 1, and take the product. (you want to say "probability", not "odds", if, in fact, those are probabilities..) –  David Mitra Nov 26 '11 at 18:59
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1 Answer

up vote 4 down vote accepted

Whenever you need to find the probability of at least one thing happening, you can instead ask "What is the probability that none of them happen?" and subtract from $1$ (since the complementary event to "none happen" is "at least one happens").

For each of the individual events, we find the probability it does not happen by subtracting the probability that it does happen from 1. We have

  • Event 1 doesn't happen: $19/21$
  • Event 2 doesn't happen: $9/10$
  • Event 3 doesn't happen: $8/15$
  • Event 4 doesn't happen: $7/16$
  • Event 5 doesn't happen: $7/10$

The probability of all these new events occurring simultaneously is just their product, which is $133/1000$.

Based on the calculation above $$ \Pr(\text{at least one event}) = 1 - \Pr(\text{none of the events}) = 1 - \frac{133}{1000} = \frac{867}{1000} = 86.7\%. $$

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