There are three coins. One is a two-headed co | vedclass

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

Question

(A) Let $E_{1}$,$E_{2}$,and $E_{3}$ be the events of choosing a two-headed coin,a biased coin,and an unbiased coin,respectively.Since one coin is chos

Vedclass · Accepted Answer

$\frac{4}{9}$

Vedclass · Answer

(A) Let $E_{1}$,$E_{2}$,and $E_{3}$ be the events of choosing a two-headed coin,a biased coin,and an unbiased coin,respectively.Since one coin is chosen at random,$P(E_{1}) = P(E_{2}) = P(E_{3}) = \frac{1}{3}$.Let $A$ be the event that the coin shows heads.For a two-headed coin,$P(A|E_{1}) = 1$.For a biased coin,$P(A|E_{2}) = 75\% = \frac{75}{100} = \frac{3}{4}$.For an unbiased coin,$P(A|E_{3}) = \frac{1}{2}$.Using Bayes' theorem,the probability that the coin is two-headed given that it shows heads is $P(E_{1}|A) = \frac{P(E_{1})P(A|E_{1})}{P(E_{1})P(A|E_{1}) + P(E_{2})P(A|E_{2}) + P(E_{3})P(A|E_{3})}$.Substituting the values: $P(E_{1}|A) = \frac{\frac{1}{3} 	imes 1}{\frac{1}{3} 	imes 1 + \frac{1}{3} 	imes \frac{3}{4} + \frac{1}{3} 	imes \frac{1}{2}}$.$P(E_{1}|A) = \frac{\frac{1}{3}}{\frac{1}{3} (1 + \frac{3}{4} + \frac{1}{2})} = \frac{1}{\frac{4+3+2}{4}} = \frac{1}{\frac{9}{4}} = \frac{4}{9}$.

Similar Questions

$A$ card from a pack of $52$ cards is lost. From the remaining $51$ cards,$n$ cards are drawn and are found to be spades. If the probability of the lost card being a spade is $\frac{11}{50}$,then $n$ is equal to

$A$ bag $P$ contains $5$ white marbles and $3$ black marbles. Four marbles are drawn at random from $P$ and are put in an empty bag $Q$. If a marble drawn at random from $Q$ is found to be black,then the probability that all the three black marbles in $P$ were transferred to the bag $Q$ is:

$A$ man is known to speak the truth $3$ out of $4$ times. He throws a die and reports that it is a six. The probability that it is actually a six,is

$A$ bag contains $(N+1)$ coins: $N$ fair coins,and one coin with 'Head' on both sides. $A$ coin is selected at random and tossed. If the probability of getting 'Head' is $\frac{9}{16}$,then $N$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module