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

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14.Probability

medium

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

Option A

Option B

Option C

Option D

Solution

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

Standard 11

Mathematics

Let $A$ and $B$ be independent events such that $\mathrm{P}(\mathrm{A})=\mathrm{p}, \mathrm{P}(\mathrm{B})=2 \mathrm{p} .$ The largest value of $\mathrm{p}$, for which $\mathrm{P}$ (exactly one of $\mathrm{A}, \mathrm{B}$ occurs $)=\frac{5}{9}$, is :

hard

(JEE MAIN-2021)

From the employees of a company, $5$ persons are selected to represent them in the managing committee of the company. Particulars of five persons are as follows :

S.No. Name Sex Age in years

$1.$ Harish $M$ $30$

$2.$ Rohan $M$ $33$

$3.$ Sheetal $F$ $46$

$4.$ Alis $F$ $28$

$5.$ Salim $M$ $41$

A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over $35$ years?

medium

Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happens is $\frac{1}{{12}}$ and the probability that neither $E$ nor $F$ happens is $\frac{1}{2},$ then

hard

(IIT-1993)

Let $A$ and $B$ be independent events with $P(A)=0.3$ and $P(B)=0.4$. Find $P(A \cap B)$

easy

Two persons $A$ and $B$ throw a (fair)die (six-faced cube with faces numbered from $1$ to $6$ ) alternately, starting with $A$. The first person to get an outcome different from the previous one by the opponent wins. The probability that $B$ wins is

normal

(KVPY-2014)

S.No.	Name	Sex	Age in years
$1.$	Harish	$M$	$30$
$2.$	Rohan	$M$	$33$
$3.$	Sheetal	$F$	$46$
$4.$	Alis	$F$	$28$
$5.$	Salim	$M$	$41$