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I struggle with finding an efficient way to notate the following tasks. From $n$ events $A_1, A_2, \dots, A_n$

  1. exactly one event happens
  2. exactly $n-1$ happens

There is a post on finding the notation for two events but not on the general case of $n$ (to the best of my knowledge).

If possible, please elaborate a bit on your solution. Thanks for any advice!

Asaf Karagila
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DavidB
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1 Answers1

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Let $\overline{A_k}$ be the complement of $A_k$, that is the event that $A_k$ does not happen. To reduce the notational clutter, notate intersection by juxtaposition. That is, $AB$ means $A\cap B$. Then the probability that exactly one event happens is $$\bigcup_{k=1}^n\overline{A_1}\overline{A_2}\cdots\overline{A_{k-1}}A_k\overline{A_{k+1}}\cdots\overline{A_n}$$

The second case may be written similarly, by reversing the complemented and non-complemented symbols.

I don't know what kind of elaboration would be appropriate. Please ask me if you have questions.

saulspatz
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  • Thank you @saulspatz! Let me ask you a follow-up question on your answer. If $A_k$ is the event that happens while the others jointly do not happen, what is the role of the $\bigcup$? What do we need it for? – DavidB Feb 06 '21 at 22:43