JEE Main & Advanced Mathematics Probability Question Bank Self Evaluation Test - Probability-I

If A and B are two events. The probability that at most one of A, B occurs, is

A) \[1-P(A\cap B)\]

B) \[P(\bar{A})+P(\bar{B})-P(\bar{A}\cap \bar{B})\]

C) \[P(\bar{A})+P(\bar{B})+P(A\cup B)-1\]

D) All of these

Correct Answer: D

Solution :

[d] Required probability \[=P(\bar{A}\cup \bar{B})=P(\overline{A\cap B})\]

\[=1-P(A\cap B)\]

Again, \[P(\bar{A}\cup \bar{B})=P(\bar{A})+P(\bar{B})-P(\bar{A}\cap \bar{B})\]

[By add. Theorem]

Again, \[P(\bar{A}\cup \bar{B})=P(\bar{A})+P(\bar{B})-P(\bar{A}\cap \bar{B})\]

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

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

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

Finally,

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

\[=P(A\cap \bar{B})+P(\bar{A}\cap B)+P(\bar{A}\cap \bar{B})\]

[\[\because A\cap \bar{B},\bar{A}\cap B\] and \[\bar{A}\cap \bar{B}\] are mutually exclusive events]

So, alternative [d] is the correct answer.