Bayes' Theorem

$\mathrm{A}$ and $\mathrm{B}$ are two students. Their chances of solving a problem correctly are $\frac{1}{3}$ and $\frac{1}{4}$, respectively. If the probability of their making a common error is $\frac{1}{20}$ and they obtain the same answer, then the probability of their answer to be correct is

An item is manufactured by three machines $A, B$ and $C$. Out of the total number of items manufactured during a specified period, $50\%$ are manufactured on $\mathrm{A},30\%$ on $B$ and $20\%$ on $C$. $2\%$ of the items produced on $A$ and $2\%$ of items produced on $B$ and $3\%$ of items produced on $C$ are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine $A$.

By examining the chest $X$ ray, the probability that TB is detected when a person is actually suffering is $0.99.$ The probability of a healthy person diagnosed to have TB is $0.001$. In a certain city, $1$ in $1000$  people suffers from TB. A person is selected at random and is diagnosed to have TB. What is the probability that he actually has TB?

There are three urns containing $2$ white and $3$ black balls, $3$ white and $2$ black balls, and $4$ white and $1$ black balls, respectively. There is an equal probability of each urn being chosen. A ball is drawn at random from the chosen urn and it is found to be white. Find the probability that the ball drawn was from the second urn.

There are two bags, one of which contains $3$ black and $4$ white balls while the other contains $4$ black and $3$ white balls. A die is thrown. If it shows up 1 or $3$, a ball is taken from the $1$st bags but it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a black ball.

A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the
envelope, just two consecutive letter TA are visible. What is the probability that the letter came from TATA NAGAR.

 A shopkeeper sells three types of flower seeds $A_1,A_2$ and $A_3$. They are sold as a mixture where the proportions are $4: 4: 2$ respectively. The germination rates of the three types of seeds are $45\%,60\%$ and $35\%$. Calculate the probability that it is of the type $A_2$ given that a randomly chosen seed does not germinate (correct up to three decimal places).

 A shopkeeper sells three types of flower seeds $A_1,A_2$ and $A_3$. They are sold as a mixture where the proportions are $4: 4: 2$ respectively. The germination rates of the three types of seeds are $45\%,60\%$ and $35\%$. Calculate the probability of a randomly chosen seed to germinate.

Three bags contain a number of red and white balls as follows:

Bag $1:3$  red balls, Bag$2:2$ red balls and $1$ white ball and

Bag $3:3$ white balls.

The probability that bag $i$ will be chosen and a ball is selected from it is $\frac{i}{6}, i=1,2,3.$ If a white ball is selected, what is the probability that it came from Bag $3$?

Three bags contain a number of red and white balls as follows:

Bag $1:3$  red balls, Bag$2:2$ red balls and $1$ white ball and Bag $3:3$ white balls. The probability that bag $i$ will be chosen and a ball is selected from it is $\frac{i}{6},i=1,2,3.$ If a white ball is selected, what is the probability that it came from Bag $2$?

Suppose that $6\%$ of the people with blood group $O$ are left handed and $10\%$ of those with other blood groups are left handed. $30\%$ of the people have blood group $O$. If a left handed person is selected at random, what is the probability that he$/$she will have blood group $O$?

Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is two headed, if you take one out, toss it and get a head, what is the probability that it was a fair coin?