JEE Main & Advanced Mathematics Probability Addition Theorems on Probability

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) \[P(A)\le P(B)\]

 

 

• Probability of occurrence of neither A nor B is

 

 

\[P(\bar{A}\cap \bar{B})=P(\overline{A\cup B})=1-P(A\cup B)\]

 

 

(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.