(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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(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
$2/5$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

Probability

Subtopic

Baye's theorem

$A$ survey of people in a given region showed that $20 \%$ were smokers. The probability of death due to lung cancer given that a person smoked was $10$ times the probability of death due to lung cancer,given that a person did not smoke. If the probability of death due to lung cancer in the region is $0.006$,what is the probability of death due to lung cancer given that a person is a smoker?

MediumWBJEE 2015

View Solution

Bag $1$ contains $4$ white balls and $5$ black balls,and Bag $2$ contains $n$ white balls and $3$ black balls. One ball is drawn randomly from Bag $1$ and transferred to Bag $2$. $A$ ball is then drawn randomly from Bag $2$. If the probability that the ball drawn from Bag $2$ is white is $29/45$,then $n$ is equal to:

DifficultJEE MAIN 2025

View Solution

There are $10$ coins in a box,out of which $8$ are normal and the remaining are with heads on both sides. $A$ coin is chosen at random from the box and tossed $6$ times. If it shows heads each time,then the probability that the selected coin has heads on both sides is

MediumTS EAMCET 2023

View Solution

Given three identical boxes $I$,$II$ and $III$,each containing two coins. In box $I$,both coins are gold coins,in box $II$,both are silver coins and in the box $III$,there is one gold and one silver coin. $A$ person chooses a box at random and takes out a coin. If the coin is of gold,what is the probability that the other coin in the box is also of gold?

Medium

View Solution

There are two balls in an urn. Each ball can be either white or black. If a white ball is put into the urn and thereafter a ball is drawn at random from the urn,then the probability that it is white is

DifficultAIEEE 2012

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

Similar Questions

There are $10$ coins in a box,out of which $8$ are normal and the remaining are with heads on both sides. $A$ coin is chosen at random from the box and tossed $6$ times. If it shows heads each time,then the probability that the selected coin has heads on both sides is

There are two balls in an urn. Each ball can be either white or black. If a white ball is put into the urn and thereafter a ball is drawn at random from the urn,then the probability that it is white is

Vedclass Test Series

Exam Paper Generator

Online Exam Module