(A) Let $P(A) = \frac{3}{4}, P(B) = \frac{1}{2}, P(C) = \frac{5}{8}$.Since they hit independently,$P(\bar{C}) = 1 - P(C) = 1 - \frac{5}{8} = \frac{3}{

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(A) Let $P(A) = \frac{3}{4}, P(B) = \frac{1}{2}, P(C) = \frac{5}{8}$.Since they hit independently,$P(\bar{C}) = 1 - P(C) = 1 - \frac{5}{8} = \frac{3}{8}$.The event that the target is hit by $A$ or $B$ but not by $C$ is $(A \cup B) \cap \bar{C}$.Since the events are independent,$P((A \cup B) \cap \bar{C}) = P(A \cup B) \times P(\bar{C})$.$P(A \cup B) = P(A) + P(B) - P(A \cap B) = P(A) + P(B) - P(A)P(B)$.$P(A \cup B) = \frac{3}{4} + \frac{1}{2} - (\frac{3}{4} \times \frac{1}{2}) = \frac{6+4-3}{8} = \frac{7}{8}$.Therefore,$P((A \cup B) \cap \bar{C}) = \frac{7}{8} \times \frac{3}{8} = \frac{21}{64}$.

Class 12 Mathematics Probability Mix Examples-Probability

Medium

$A, B, C$ try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are $\frac{3}{4}, \frac{1}{2}, \frac{5}{8}$. The probability that the target is hit by $A$ or $B$ but not by $C$ is

JEE MAIN 2013 All JEE MAIN

A
$21/64$
B
$7/8$
C
$7/32$
D
$9/64$

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Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die,then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is:

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$E_1$ and $E_2$ are two independent events of a random experiment with $P(E_1) = \frac{1}{2}$ and $P(E_1 \cup E_2) = \frac{2}{3}$. Match the items of List-$I$ with those of List-$II$.
List-$I$ List-$II$
$A. P(E_2) =$ $I. 2/3$
$B. P(E_1 | E_2) =$ $II. 5/6$
$C. P(\bar{E}_2 | E_1) =$ $III. 1/3$
$D. P(\bar{E}_1 \cup \bar{E}_2) =$ $IV. 1/2$

MediumTS EAMCET 2020

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$A$ man throws a fair coin repeatedly. He gets $10$ points for each head he throws and $5$ points for each tail he throws. If the probability that he gets exactly $30$ points is $\frac{m}{n}$, where $\text{gcd}(m, n) = 1$, then $m + n$ is equal to:

DifficultJEE MAIN 2026

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If $A$ and $B$ are independent events of a random experiment such that $P(A \cap B) = \frac{1}{6}$ and $P(\bar{A} \cap \bar{B}) = \frac{1}{3}$,then $P(A)$ is equal to (Here,$\bar{E}$ is the complement of the event $E$)

DifficultTS EAMCET 2008

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Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die,then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is

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List-$I$	List-$II$
$A. P(E_2) =$	$I. 2/3$
$B. P(E_1 \| E_2) =$	$II. 5/6$
$C. P(\bar{E}_2 \| E_1) =$	$III. 1/3$
$D. P(\bar{E}_1 \cup \bar{E}_2) =$	$IV. 1/2$

$A, B, C$ try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are $\frac{3}{4}, \frac{1}{2}, \frac{5}{8}$. The probability that the target is hit by $A$ or $B$ but not by $C$ is

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Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die,then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is:

$A$ man throws a fair coin repeatedly. He gets $10$ points for each head he throws and $5$ points for each tail he throws. If the probability that he gets exactly $30$ points is $\frac{m}{n}$, where $\text{gcd}(m, n) = 1$, then $m + n$ is equal to:

If $A$ and $B$ are independent events of a random experiment such that $P(A \cap B) = \frac{1}{6}$ and $P(\bar{A} \cap \bar{B}) = \frac{1}{3}$,then $P(A)$ is equal to (Here,$\bar{E}$ is the complement of the event $E$)

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die,then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is

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