(A) Let $E_A$ be the event that $A$ fails and $E_B$ be the event that $B$ fails.Given:$P(E_A) = 0.2$$P(E_A \cap E_B) = 0.15$$P(B \text{ fails alone})

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(A) Let $E_A$ be the event that $A$ fails and $E_B$ be the event that $B$ fails.Given:$P(E_A) = 0.2$$P(E_A \cap E_B) = 0.15$$P(B \text{ fails alone}) = P(E_B) - P(E_A \cap E_B) = 0.15$Substituting the values:$P(E_B) - 0.15 = 0.15$$P(E_B) = 0.3$Now,the conditional probability $P(E_A | E_B)$ is given by:$P(E_A | E_B) = \frac{P(E_A \cap E_B)}{P(E_B)}$$P(E_A | E_B) = \frac{0.15}{0.3} = 0.5$.

Class 12 Mathematics Probability Conditional probability

Easy

An electronic assembly consists of two subsystems,$A$ and $B$. From previous testing procedures,the following probabilities are known:
$P(A \text{ fails}) = 0.2$
$P(B \text{ fails alone}) = 0.15$
$P(A \text{ and } B \text{ fail}) = 0.15$
Evaluate the probability $P(A \text{ fails } | \text{ } B \text{ has failed})$.

A
$0.5$
B
$0.4$
C
$0.3$
D
$0.2$

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Class 12

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Probability

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Conditional probability

$A$ fair die is rolled. Consider events $E=\{1,3,5\}, F=\{2,3\},$ and $G=\{2,3,4,5\}$. Find $P(E | G)$ and $P(G | E)$.

Medium

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Let $A$ and $B$ be two events with $P(A^{C}) = 0.3$,$P(B) = 0.4$,and $P(A \cap B^{C}) = 0.5$. Then $P(B \mid A \cup B^{C})$ is equal to

EasyWBJEE 2012

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If $A$ and $B$ are events,such that $P(A) = \frac{1}{4}$,$P(A|B) = \frac{1}{2}$,and $P(B|A) = \frac{2}{3}$,then $P(B)$ is

EasyKCET 2023

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In a city,$40\%$ of the people have brown hair,$25\%$ have brown eyes,and $15\%$ have both brown hair and brown eyes. If a person is selected at random from those having brown hair,what is the probability that they also have brown eyes?

Medium

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Suppose $A$ and $B$ are events of a random experiment such that $P(A)=\frac{1}{3}$,$P(A \cap B)=\frac{1}{5}$ and $P(A \cup B)=\frac{3}{5}$. Match the items of List-$I$ with the items of List-$II$.
List-$I$ List-$II$
$A$. $P(\frac{A}{B})$ $(i)$. $\frac{2}{15}$
$B$. $P(\bar{B})$ $(ii)$. $\frac{4}{15}$
$C$. $P(A \cap \bar{B})$ $(iii)$. $\frac{8}{15}$
$D$. $P(B \cap \bar{A})$ $(iv)$. $\frac{2}{3}$
$(v)$. $\frac{3}{7}$

MediumTS EAMCET 2018

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List-$I$	List-$II$
$A$. $P(\frac{A}{B})$	$(i)$. $\frac{2}{15}$
$B$. $P(\bar{B})$	$(ii)$. $\frac{4}{15}$
$C$. $P(A \cap \bar{B})$	$(iii)$. $\frac{8}{15}$
$D$. $P(B \cap \bar{A})$	$(iv)$. $\frac{2}{3}$
	$(v)$. $\frac{3}{7}$

Similar Questions

$A$ fair die is rolled. Consider events $E=\{1,3,5\}, F=\{2,3\},$ and $G=\{2,3,4,5\}$. Find $P(E | G)$ and $P(G | E)$.

Let $A$ and $B$ be two events with $P(A^{C}) = 0.3$,$P(B) = 0.4$,and $P(A \cap B^{C}) = 0.5$. Then $P(B \mid A \cup B^{C})$ is equal to

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