Addition Theorems on Probability
Category : JEE Main & Advanced
Notations : (i) \[P(A+B)\text{ or }P(A\cup B)=\] Probability of happening of A or B
= Probability of happening of the events A or B or both
= Probability of occurrence of at least one event A or B
(ii) \[P(AB)\] or \[P(A\cap B)=\] Probability of happening of events A and B together.
(1) When events are not mutually exclusive : If A and B are two events which are not mutually exclusive, then
\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]
or \[P(A+B)=P(A)+P(B)-P(AB)\]
For any three events A, B, C
\[P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)\]\[-P(B\cap C)-P(C\cap A)+P(A\cap B\cap C)\]
or \[P(A+B+C)=P(A)+P(B)+P(C)-P(AB)-P(BC)\]\[-P(CA)+P(ABC)\]
(2) When events are mutually exclusive : If A and B are mutually exclusive events, then \[n(A\cap B)=0\] \[\Rightarrow \] \[P(A\cap B)=0\]
\[\therefore \]\[P(A\cup B)=P(A)+P(B)\].
For any three events A, B, C which are mutually exclusive,
\[P(A\cap B)=P(B\cap C)=P(C\cap A)=P(A\cap B\cap C)=0\]
\[\therefore \]\[P(A\cup B\cup C)=P(A)+P(B)+P(C)\].
The probability of happening of any one of several mutually exclusive events is equal to the sum of their probabilities, i.e. if \[{{A}_{1}},\,{{A}_{2}}.....{{A}_{n}}\] are mutually exclusive events, then
\[P({{A}_{1}}+{{A}_{2}}+...+{{A}_{n}})=P({{A}_{1}})+P({{A}_{2}})+.....+P({{A}_{n}})\]
i.e. \[P(\sum{{{A}_{i}}})=\sum{P({{A}_{i}})}\].
(3) When events are independent : If A and B are independent events, then \[P(A\cap B)=P(A).P(B)\]
\[\therefore \] \[P(A\cup B)=P(A)+P(B)-P(A).P(B)\].
(4) Some other theorems
(i) Let A and B be two events associated with a random experiment, then
(a) \[P(\bar{A}\cap B)=P(B)-P(A\cap B)\]
(b) \[P(A\cap \bar{B})=P(A)-P(A\cap B)\]
If \[B\subset A,\] then
(a) \[P(A\cap \bar{B})=P(A)-P(B)\]
(b) \[P(B)\le P(A)\]
Similarly if \[A\subset B,\] then
(a) \[(\bar{A}\cap B)=P(B)-P(A)\]
(b) \[P(A)\le P(B)\]
\[P(\bar{A}\cap \bar{B})=P(\overline{A\cup B})=1-P(A\cup B)\]
(ii) Generalization of the addition theorem : If \[{{A}_{1}},\,{{A}_{2}},.....,\,{{A}_{n}}\] are \[n\] events associated with a random experiment, then \[P\left( \bigcup\limits_{i=1}^{n}{{{A}_{i}}} \right)=\sum\limits_{i=1}^{n}{P({{A}_{i}})}-\sum\limits_{\begin{smallmatrix} i,\,j=1 \\\,i\ne j\end{smallmatrix}}^{n}{P({{A}_{i}}\cap {{A}_{j}})}+\sum\limits_{\begin{smallmatrix} i,\,j,\,k=1 \\\,i\ne j\ne k\end{smallmatrix}}^{n}{P({{A}_{i}}\cap {{A}_{j}}\cap {{A}_{k}})}+\]\[...+{{(-1)}^{n-1}}P({{A}_{1}}\cap {{A}_{2}}\cap .....\cap {{A}_{n}})\].
If all the events \[{{A}_{i}}(i=1,\,2...,\,n)\] are mutually exclusive, then \[P\,\,\left( \bigcup\limits_{i=1}^{n}{{{A}_{i}}} \right)=\sum\limits_{i=1}^{n}{P({{A}_{i}})}\]
i.e. \[P({{A}_{1}}\cup {{A}_{2}}\cup ....\cup {{A}_{n}})=P({{A}_{1}})+P({{A}_{2}})+....+P({{A}_{n}})\].
(iii) Booley’s inequality : If \[{{A}_{1}},\,{{A}_{2}},\,....{{A}_{n}}\] are n events associated with a random experiment, then
(a) \[P\left( \bigcap\limits_{i=1}^{n}{{{A}_{i}}} \right)\ge \sum\limits_{i=1}^{n}{P({{A}_{i}})-(n-1)}\]
(b) \[P\left( \bigcup\limits_{i=1}^{n}{{{A}_{i}}} \right)\le \sum\limits_{i=1}^{n}{P({{A}_{i}})}\]
These results can be easily established by using the Principle of mathematical induction.
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