Prove that if $E$ and $F$ are independent events, then so are the events $\mathrm{E}$ and $\mathrm{F}^{\prime}$.

Vedclass pdf generator app on play store
Vedclass iOS app on app store

since $\mathrm{E}$ and $\mathrm{F}$ are independent, we have

$\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})$       ......... $(1)$

From the venn diagram in Fig it is clear that $E \cap \mathrm{F}$ and $\mathrm{E} \cap \mathrm{F}^{\prime}$ are mutually exclusive events and also $\mathrm{E}=(\mathrm{E} \cap \mathrm{F}) \cup\left(\mathrm{E} \cap \mathrm{F}^{\prime}\right)$

Therefore        $\quad P(E)=P(E \cap F)+P\left(E \cap F^{\prime}\right)$

or                   $P\left(E \cap F^{\prime}\right)=P(E)-P(E \cap F)$

                    $=\mathrm{P}(\mathrm{E})-\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})$    (by $(1))$

                   $=\mathrm{P}(\mathrm{E})(1-\mathrm{P}(\mathrm{F}))$

                   $=\mathrm{P}(\mathrm{E})$ . $\mathrm{P}\left(\mathrm{F}^{\prime}\right)$

Hence, $\mathrm{E}$ and $\mathrm{F}^{\prime}$ are independent

863-s41

Similar Questions

One card is drawn from a pack of $52$ cards. The probability that it is a queen or heart is

Fill in the blanks in following table :

$P(A)$ $P(B)$ $P(A \cap B)$ $P (A \cup B)$
$0.5$ $0.35$ .........  $0.7$

In class $XI$ of a school $40\%$ of the students study Mathematics and $30 \%$ study Biology. $10 \%$ of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology.

For any two events $A$ and $B$ in a sample space

  • [IIT 1991]

In a hostel, $60 \%$ of the students read Hindi newspaper, $40 \%$ read English newspaper and $20 \%$ read both Hindi and English newspapers. A student is selected at random Find the probability that she reads neither Hindi nor English newspapers.