(B) Let $E_{1}$ and $E_{2}$ be the events that the item is produced by machines $A$ and $B$ respectively. Let $X$ be the event that the item is defect

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(B) Let $E_{1}$ and $E_{2}$ be the events that the item is produced by machines $A$ and $B$ respectively. Let $X$ be the event that the item is defective.Given:$P(E_{1}) = 60 \% = 60/100 = 3/5$$P(E_{2}) = 40 \% = 40/100 = 2/5$$P(X | E_{1}) = 2 \% = 2/100$$P(X | E_{2}) = 1 \% = 1/100$We need to find $P(E_{2} | X)$. By Bayes' theorem:$P(E_{2} | X) = \frac{P(E_{2}) \cdot P(X | E_{2})}{P(E_{1}) \cdot P(X | E_{1}) + P(E_{2}) \cdot P(X | E_{2})}$Substituting the values:$P(E_{2} | X) = \frac{(2/5) \cdot (1/100)}{(3/5) \cdot (2/100) + (2/5) \cdot (1/100)}$$P(E_{2} | X) = \frac{2/500}{6/500 + 2/500}$$P(E_{2} | X) = \frac{2/500}{8/500}$$P(E_{2} | X) = 2/8 = 1/4$

Class 12 Mathematics Probability Baye's theorem

Medium

$A$ factory has two machines $A$ and $B$. Past record shows that machine $A$ produced $60 \%$ of the items of output and machine $B$ produced $40 \%$ of the items. Further,$2 \%$ of the items produced by machine $A$ and $1 \%$ produced by machine $B$ were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine $B$?

A
$1/5$
B
$1/4$
C
$1/3$
D
$2/5$

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Class 12

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Probability

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Baye's theorem

Bag $I$ contains $3$ red and $2$ green balls and Bag $II$ contains $5$ red and $3$ green balls. $A$ ball is drawn from one of the bags at random and it is found to be green. Then the probability that it is drawn from Bag $I$ is

MediumMHT CET 2025

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There are two dice $A$ and $B$. Die $A$ has $4$ red and $2$ white faces,and die $B$ has $2$ red and $4$ white faces. $A$ coin is tossed once. If it shows head,die $A$ is rolled; if it shows tail,die $B$ is rolled. If the probability that die $A$ is used is $\left(\frac{32}{33}\right)$ given that red turns up every time in the first $n$ throws,then $n=$

MediumAP EAMCET 2020

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$70 \%$ of the total employees of a factory are men. Among the employees of that factory,$30 \%$ of men and $15 \%$ of women are technical assistants. If an employee chosen at random is found to be a technical assistant,then the probability that this employee is a man is

MediumAP EAMCET 2025

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There are two boxes, each containing $10$ balls. In each box, some are black and the rest are white. $A$ ball is drawn at random from one of the boxes and it is found to be black. If the probability that the black ball drawn is from the second box is $\frac{1}{5}$, then the number of black balls in the first box is:

MediumTS EAMCET 2025

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Bag $A$ contains $3$ white and $5$ black balls, while bag $B$ contains $4$ white and $3$ black balls. $A$ ball is selected at random from bag $A$ and put into bag $B$. If a ball is now selected at random from bag $B$, what is the probability that this ball is white?

DifficultMHT CET 2026

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Bag $I$ contains $3$ red and $2$ green balls and Bag $II$ contains $5$ red and $3$ green balls. $A$ ball is drawn from one of the bags at random and it is found to be green. Then the probability that it is drawn from Bag $I$ is

$70 \%$ of the total employees of a factory are men. Among the employees of that factory,$30 \%$ of men and $15 \%$ of women are technical assistants. If an employee chosen at random is found to be a technical assistant,then the probability that this employee is a man is

Bag $A$ contains $3$ white and $5$ black balls, while bag $B$ contains $4$ white and $3$ black balls. $A$ ball is selected at random from bag $A$ and put into bag $B$. If a ball is now selected at random from bag $B$, what is the probability that this ball is white?

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