Bayes' Theorem

A first bag contains five white balls and ten black balls and the second bag contains six white balls and four black balls. The experiment consists of selecting a bag and then drawing a ball from the selected bag. Find the probability of drawing a white ball.

In a bolt factory, machines $A,B$ and $C$ manufacture respectively $20\%, 30\%$ and  $50\%$ of the total bolts. Of their output $3, 4$ and $2$ percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective then the probability that it is manufactured by the machine $C$ is 

A bolt manufacturing company has three products $A, B$ and $C$. $50\% \& 30\%$ of the products are $A$ and $B$ type respectively and remaining are $C$ type. Then, the probability that the product $A$ is defective is $4\%$, that $B$ is defective is $3\%$ and that $C$ is defective is $2\%$. A product is picked randomly and found to be defective, then the probability that it is type  $C$ is

A box has $5$ balls. Three balls are drawn at random and are found to be white. What is the probability that the box has all the white balls?

In a class of $75$ students, $15$ are above average, $45$ are average and rest are below average achievers. The probability that an above average achieving student fails is $0.005$, that an average achieving student fails is $0.05$ and the probability of a below average achieving student failing is $0.15$. If a student is known to have passed, what is the probability that he is a below average achiever?

One bag contains $3$ red and $5$ black balls. Another bag contains $6$ red balls and $4$ black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is red.

It is estimated that $50\%$ of emails are spam emails. Some software has been applied to filter these spam emails before they reach your inbox. A certain brand of software claims that it can detect $99\%$ of spam emails, and the probability for a false positive (a non-spam email detected as spam) is $5\%.$

If an email is detected as spam, then what is the probability that it is in fact a non-spam email?

Bhavani is going to play a game of chess against one of four opponents in an intercollege sports competition. Each opponent is equally likely to be paired against her. The table below shows the chances of Bhavani losing, when paired against each opponent.

If the probability that Bhavani loses the game that day is $\frac{1}{2}$, find the probability for Bhavani to be losing the game when paired against Opponent $3$. Show your steps.

From a pack of $52$ playing cards, a card is lost. From the remaining $51$ cards, two cards are drawn at random $($without replacement$)$ and are both found to be diamonds. What is the probability that the lost card was a card of heart?

If a machine is correctly set up, it produces $90\%$ acceptable items. If it is incorrectly set up, it produces only $40\%$ acceptable items. Past experience shows that $80\%$ of the set ups are correctly done. If after a certain set up, the machine produces $2$ acceptable items, find the probability that the machine is correctly setup.

Coloured balls are distributed in four boxes as shown in the following table:

A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III?

Suppose that the reliability of a $HIV$ test is specified as follows: Of people having $HIV$, $90\%$ of the test detect the disease but $10\%$ go undetected. Of people free of $HIV, 99\%$ of the test are judged$HIV-ive$ but $1\%$ are diagnosed as showing $HIV+ive$. From a large population of which only $0.1\%$ have $HIV$, one person is selected at random, given the$HIV$ test, and the pathologist reports him/her as $HIV+ive$. What is the probability that the person actually has $HIV$?

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?

Bag $I$ contains $3$ red and $4$ black balls while another Bag $II$ contains $5$ red and $6$ black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag $II$.

A person has undertaken a construction job. The probabilities are $0.65$ that there will be strike, $0.80$ that the construction job will be completed on time if there is no strike, and $0.32$ that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time.

A person draws two cards successively without replacement from a pack of $52$ cards. He tells that both cards are aces, what is the probability that both are aces if there are $60\%$ chances that he speaks truth?

An artillery target may be either at point $\mathrm{I}$ with probability $\frac{8}{9}$ or at the point $\mathrm{II}$ with probability $\frac{1}{9}.$ There are $21$ shells each of which can be fired either at point $\mathrm{I}$ or $\mathrm{II}.$ Each shell may hit the target independently of the other shell with probability $\frac{1}{2}.$ Minimum number of shells that must be fired at point $\mathrm{I}$ to hit the target with maximum probability is equal to $k$ then $\frac{k}{3}=$

A bag $P$ contains $4$ white and $4$ red balls and a second bag $Q$ contains $2$ white balls and $6$ red balls. One ball is drawn at random from one of the bags and it is found to be white. The probability that the ball was drawn from the bag $P$ is

If a machine is set up correctly, it produces $90\%$ acceptable items. If it is incorrectly set, it produces only $30\%$ acceptable items. Past experience shows that $80\%$ of the set ups are correctly done. If after a certain set up, the machine produces $2$ acceptable items, the probability that the machine was correctly set up is

In a university, 30% of the students are doing a course in statistics use the book authored by ${\mathrm{A}}_1$,45% use the book authored by ${\mathrm{A}}_2$ and 25% use the book authored by ${\mathrm{A}}_3$. The proportion of the students who learnt about each of these books through their teacher are $\mathrm{P}\left({\mathrm{A}}_1\right)=0.50,\mathrm{P}\left({\mathrm{A}}_2\right)=0.30 \mathrm{and} \mathrm{P}\left({\mathrm{A}}_3\right)=0.20$. One of the students selected at random revealed that he learned the book he is using through their teachers. Find the probability that the book used it was authored by ${\mathrm{A}}_1,{\mathrm{A}}_2 \mathrm{and} {\mathrm{A}}_3$ respectively.