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christang_1023

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Now I wonder if there is the more general form of addition law, which applies to any n elements.

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- Thread starter christang_1023
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In summary, there are two forms of the addition law in probability, one with two elements and one with three elements. However, there is also a more general form known as the principle of inclusion-exclusion, which can be applied to any number of elements. This principle is often discussed in probability books and can be derived using indicator random variables.

- #1

christang_1023

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Now I wonder if there is the more general form of addition law, which applies to any n elements.

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S.G. Janssens

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Yes, there is. It is knows as the principle of inclusion-exclusion, usually discussed in probability books and also in various places online.christang_1023 said:Now I wonder if there is the more general form of addition law, which applies to any n elements.

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StoneTemplePython

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i) ##P\Big( A_1 \bigcup A_2 \bigcup A_3 \bigcup... \bigcup A_n\Big) = 1- P\Big( A_1^C \bigcap A_2^C \bigcap A_3^C \bigcap ... \bigcap A_n^C\Big)##

i.e. recall that the union of events is equivalent to the complement of the intersection of complements

ii) then use indicator random variables to model the RHS, which gives

##P\Big( A_1 \bigcup A_2 \bigcup A_3 \bigcup ... \bigcup A_n\Big) = 1 - E\Big[\big(1-\mathbb I_{A_1}\big)\big(1-\mathbb I_{A_2}\big)\big(1-\mathbb I_{A_3}\big)...\big(1-\mathbb I_{A_n}\big)\Big]##

if you expand the polynomial (i.e. apply elementary symmetric functions of the indicators and pay attention to the sign), and then use linearity of the expectations operator, you recover Inclusion-Exclusion for probability.

The addition law for probability with multiple elements states that the probability of the union of two or more events is equal to the sum of their individual probabilities, minus the probability of their intersection.

This law is used to calculate the probability of a compound event, where multiple events occur simultaneously or sequentially.

The union of two events refers to the occurrence of either one or both events. The intersection of two events refers to the occurrence of both events simultaneously.

Yes, the addition law for probability can be applied to any number of events. The formula remains the same - the sum of individual probabilities minus the probability of their intersection.

Yes, this law assumes that the events are mutually exclusive, meaning that they cannot occur at the same time. If the events are not mutually exclusive, the addition law may overestimate the probability of the compound event.

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