Probability of Occurrence of At Least One Independent Event

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Theorem

Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $A_1, A_2, \ldots, A_m \in \Sigma$ be independent events in the event space of $\EE$.

Then the probability of at least one of $A_1$ to $A_m$ occurring is:

$\ds 1 - \prod_{i \mathop = 1}^m \paren {1 - \map \Pr {A_i} }$

Corollary

Let $A$ be an event in an event space of an experiment $\EE$ whose probability space is $\struct {\Omega, \Sigma, \Pr}$.

Let $\map \Pr A = p$.

Suppose that the nature of $\EE$ is that its outcome is independent of previous trials of $\EE$.

Then the probability that $A$ occurs at least once during the course of $m$ trials of $\EE$ is $1 - \paren {1 - p}^m$.

Proof

Follows as a direct result of Probability of Independent Events Not Happening.

Let $B$ be the event "None of $A_1$ to $A_m$ happen".

From Probability of Independent Events Not Happening:

$\ds \map \Pr B = \prod_{i \mathop = 1}^m \paren {1 - \map \Pr {A_i} }$

Then $\Omega \setminus B$ is the event "Not none of $A_1$ to $A_m$ happen", or "At least one of $A_1$ to $A_m$ happens".

From Elementary Properties of Probability Measure:

$\forall A \in \Omega: \map \Pr {\Omega \setminus A} = 1 - \map \Pr A$

Hence the probability that at least one of $A_1$ to $A_m$ happen is:

$\ds 1 - \map \Pr B = 1 - \prod_{i \mathop = 1}^m \paren {1 - \map \Pr {A_i} }$

Proof of Corollary

It can immediately be seen that this is an instance of the main result with all of $A_1, A_2, \ldots, A_m$ being instances of $A$.

The result follows directly.

$\blacksquare$

Example

This is a classic result which contradicts the following equally classic fallacy:

"There is a one in six chance of throwing a six with a single throw of a die.

Therefore, there is a two in six chance of throwing a six on two throws of a die."

In fact this is an example of "occurrence of at least one independent event".

The probability of throwing at least one six on two throws of a die is in fact:

$1 - \paren {1 - \dfrac 1 6}^2 = \dfrac {11} {36} < \dfrac 2 6$

Not a lot in it, but definitely significantly less.

See De Méré's Paradox for a real-world application of this result as it occurred in history.