Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is $0.05$ and that Ashima will qualify the examination is $0.10 .$ The probability that both will qualify the examination is $0.02 .$ Find the probability that Only one of them will qualify the examination.
Let $E$ and $F$ denote the events that Anil and Ashima will qualify the examination, respectively. Given that
$P(E)=0.05$, $P(F)=0.10$ and $P(E \cap F)=0.02$
Then
The event only one of them will qualify the examination is same as the event either (Anil will qualify, andAshima will not qualify) or (Anil will not qualify and Ashima will qualify) i.e., $E \cap F ^{\prime}$ or $E ^{\prime} \cap F ,$ where $E \cap F ^{\prime}$ and $E ^{\prime} \cap F$ are mutually exclusive.
Therefore, $P$ (only one of them will qualify) $=P(E \cap F^{\prime} $ or $E^{\prime} \cap F)$
$= P \left( E \cap F ^{\prime}\right)$ $+ P \left( E ^{\prime} \cap F \right)$ $= P ( E )- P ( E \cap F )+ P ( F )- P ( E \cap F ) $
$=0.05-0.02+0.10-0.02=0.11$
A card is drawn from a pack of cards. Find the probability that the card will be a queen or a heart
If $P(B) = \frac{3}{4}$, $P(A \cap B \cap \bar C) = \frac{1}{3}{\rm{ }}$ and $P(\bar A \cap B \cap \bar C) = \frac{1}{3},$ then $P(B \cap C)$ is
If $P(A) = 0.25,\,\,P(B) = 0.50$ and $P(A \cap B) = 0.14,$ then $P(A \cap \bar B)$ is equal to
For any two events $A$ and $B$ in a sample space
Let $A$ and $B $ be two events such that $P\left( {\overline {A \cup B} } \right) = \frac{1}{6}\;,P\left( {A \cap B} \right) = \frac{1}{4}$ and $P\left( {\bar A} \right) = \frac{1}{4}$ where $\bar A$ stands for the complement of the event $A$. Then the events $A$ and$B$ are