(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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Probability

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Baye's theorem

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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In an entrance test,there are multiple-choice questions. There are four possible answers to each question,of which one is correct. The probability that a student knows the answer to a question is $9/10$. If he gets the correct answer to a question,then the probability that he was guessing is:

DifficultTS EAMCET 2012

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$A$ card from a pack of $52$ cards is lost. From the remaining cards of the pack,two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

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$A$ bag contains $8$ balls,whose colors are either white or black. $4$ balls are drawn at random without replacement and it was found that $2$ balls are white and $2$ balls are black. The probability that the bag originally contained an equal number of white and black balls is:

MediumJEE MAIN 2024

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$25 \%$ of the population are smokers. $A$ smoker has $27$ times more chances to develop lung cancer than a non-smoker. $A$ person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$. Then the value of $k$ is $.............$

DifficultJEE MAIN 2023

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In an entrance test,there are multiple-choice questions. There are four possible answers to each question,of which one is correct. The probability that a student knows the answer to a question is $9/10$. If he gets the correct answer to a question,then the probability that he was guessing is:

$A$ card from a pack of $52$ cards is lost. From the remaining cards of the pack,two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

$A$ bag contains $8$ balls,whose colors are either white or black. $4$ balls are drawn at random without replacement and it was found that $2$ balls are white and $2$ balls are black. The probability that the bag originally contained an equal number of white and black balls is:

$25 \%$ of the population are smokers. $A$ smoker has $27$ times more chances to develop lung cancer than a non-smoker. $A$ person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$. Then the value of $k$ is $.............$

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