(C) Let $K$ be the event that the student knows the answer and $G$ be the event that the student guesses the answer.Let $C$ be the event that the stud

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(C) Let $K$ be the event that the student knows the answer and $G$ be the event that the student guesses the answer.Let $C$ be the event that the student gives the correct answer.We are given:$P(C|G) = \frac{1}{2}$$P(C|K) = 1$ (since the student always gives the correct answer if he knows it).$P(G|C) = \frac{1}{6}$Let $P(K) = x$. Then $P(G) = 1 - x$.Using Bayes' theorem:$P(G|C) = \frac{P(C|G)P(G)}{P(C|G)P(G) + P(C|K)P(K)}$Substituting the values:$\frac{1}{6} = \frac{(\frac{1}{2})(1-x)}{(\frac{1}{2})(1-x) + (1)(x)}$$\frac{1}{6} = \frac{\frac{1-x}{2}}{\frac{1-x+2x}{2}}$$\frac{1}{6} = \frac{1-x}{1+x}$$1+x = 6-6x$$7x = 5$$x = \frac{5}{7}$Thus,the probability that the student knows the answer is $\frac{5}{7}$.

Class 12 Mathematics Probability Baye's theorem

$A$ student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question,he gives the correct answer. Assume that the probability of the student giving the correct answer for a question,given that he has guessed it,is $\frac{1}{2}$. Also assume that the probability of the answer for a question being guessed,given that the student's answer is correct,is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question is

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A
$\frac{1}{12}$
B
$\frac{1}{7}$
C
$\frac{5}{7}$
D
$\frac{5}{12}$

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Bag $A$ contains $3$ white and $7$ red balls,and bag $B$ contains $3$ white and $2$ red balls. One bag is selected at random and a ball is drawn from it. What is the probability that the ball was drawn from bag $A$,given that the ball drawn is white?

MediumJEE MAIN 2024

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For $k=1, 2, 3$,the box $B_k$ contains $k$ red balls and $(k+1)$ white balls. Let $P(B_1) = \frac{1}{2}$,$P(B_2) = \frac{1}{3}$,and $P(B_3) = \frac{1}{6}$. $A$ box is selected at random and a ball is drawn from it. If a red ball is drawn,then the probability that it comes from box $B_2$ is:

MediumTS EAMCET 2008

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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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$A$ manufacturer has three machine operators $A$,$B$,and $C$. The first operator $A$ produces $1 \%$ defective items,whereas the other two operators $B$ and $C$ produce $5 \%$ and $7 \%$ defective items respectively. $A$ is on the job for $50 \%$ of the time,$B$ is on the job for $30 \%$ of the time,and $C$ is on the job for $20 \%$ of the time. If a defective item is produced,what is the probability that it was produced by $A$ (in $/34$)?

Difficult

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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

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Bag $A$ contains $3$ white and $7$ red balls,and bag $B$ contains $3$ white and $2$ red balls. One bag is selected at random and a ball is drawn from it. What is the probability that the ball was drawn from bag $A$,given that the ball drawn is white?

For $k=1, 2, 3$,the box $B_k$ contains $k$ red balls and $(k+1)$ white balls. Let $P(B_1) = \frac{1}{2}$,$P(B_2) = \frac{1}{3}$,and $P(B_3) = \frac{1}{6}$. $A$ box is selected at random and a ball is drawn from it. If a red ball is drawn,then the probability that it comes from box $B_2$ is:

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:

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