Total Probability and 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.

Three companies USHA, MODI and Bajaj supply fans to a school. The percentage of fans supplied by individual companies and the probability of those fans being defectives are given below:

Given that fan is defective, find the probability that fan is from company USHA.

A lady has two bags. In the first bag, there are $6$ red balls, $3$ black balls, $5$ green balls and $4$ white balls. In the second bag, there are $8$ red balls, $4$ black balls, $6$ green balls and $4$  white balls. The lady tosses a coin and if it comes to head she goes for the first bag and the second bag for the tail. If the chosen ball is a red ball, find the probability that the red ball is from the first bag.

A box contains $5$ black and $4$ white balls. A ball is drawn at random and its colour is noted. The ball is then put back in the box along with two additional balls of its opposite colour. If a ball is drawn again from the box, then the probability that the ball drawn now is black, is

A bag contains three coins, one of which has head on both sides, another is a biased coin that shows up heads $90\%$ of the time and the third one is an unbiased coin. A coin is taken out from the bag at random and tossed. If it shows up a head, then the probability that it is the unbiased coin, is

A purse contains $4$ copper coins and $3$ silver coins while the second purse contains $6$ copper coins and $2$ silver coins. A coin is taken out from any purse. The probability that it is a copper coin is : 

A purse contains $4$ copper and $3$ silver coins. Another purse contains $6$ copper and $2$ silver coins. A coin is taken out from any purse, the probability that it is a silver coin, is

A bag contains $4$ red and $6$ black balls. A ball is drawn at random from the bag, its color is observed and this ball along with two additional balls of the same color are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is:

A teacher goes to school either by a car as by a bike as by bus. The probabilities of his using car, bike & bus respectively $\frac{1}{5}, \frac{2}{5} \& \frac{2}{5}$. The probabilities of teacher reaching school in time by car, bike & bus respectively $\frac{1}{7}, \frac{5}{7} \& \frac{1}{7}$. If teacher reaches late in the school, then the chance that he used bike is

Urn $X$ has $5$ black balls & one pink ball and Urn $Y$ has $6$ black balls. If two balls are randomly picked up from Urn $X$ and transferred to Urn $Y$ and again two balls are randomly picked up from Urn $Y$ and transferred to Urn $X$, then the probability that pink ball is now in Urn $X$ is :-

In medical diagnostics for a disease, sensitivity (denoted a) of a test refers to the probability that a test result is positive for a person with the disease whereas specificity (denoted b) refers to the probability that a person without the disease test negative. A diagnostic test for influenza has the values of $\mathrm{a}=0.9$ and $\mathrm{b}=0.9.$ Assume that the prevalence of influenza in a population in $50\%.$ If a randomly chosen person tests negative, what is the probability that the person actually has influenza?

In medical diagnostics for a disease, sensitivity (denoted by $a$) of a test refers to the probability that a test result is positive for a person with the disease, whereas specificity (denoted by $b$) refers to the probability that a person without the disease tests negative. A diagnostic test for $\mathrm{COVID}-19$ has the values of $a=0.99$ and $b=0.99$. If the prevalence of $\mathrm{COVID}-19$ in a population is estimated to be $10\%$, what is the probability that a randomly chosen person tests positive for $\mathrm{COVID}-19$ ?

The probability of men getting a certain disease is $\frac{1}{2}$ and that of women getting the same disease is $\frac{1}{5}.$ The blood test that identifies the disease gives the correct result with probability $\frac{4}{5}$. Suppose a person is chosen at random from a group of $30$ males and $20$ females, and the blood test of that person is found to be positive. What is the probability that the chosen person is a man?

Ravi and Rashmi are each holding $2$ red cards and $2$ black cards (all four red and all four black cards are identical). Ravi picks a card at random from Rashmi and then Rashmi picks a card to random from Ravi. This process is repeated a second time. Let $p$ be the probability that both have all $4$ cards of the same colour. Then, $p$ satisfies

In a certain recruitment test with multiple choice questions, there are four options to each question, out of which only one is correct. An intelligent student knows $90\%$ of the correct answers while a weak student knows only $20\%$of the correct answers. If a weak student gets the correct answer, the probability that he was guessing is

One percent of the population suffers from a certain disease. There is blood test for this disease, and it is $99\%$ accurate, in other words, the probability that it gives the correct answer is $0.99$, regardless of whether the person is sick or healthy. A person takes the blood test, and the result says that he has the disease. The probability that he actually has the disease, is -

A box contains $6$ red, $5$ blue and $4$ white marbles. Four marbles are chosen at random without replacement. The probability that there is atleast one marble of each colour among the four chosen, is -

$\mathrm{A}$ and $\mathrm{B}$ are two independent witnesses (i.e. there is no collusion between them in a case. The probability that $\mathrm{A}$ will speak the truth is $x$ and the probability that $\mathrm{B}$ will speak the truth is $y$. $\mathrm{A}$ and $\mathrm{B}$ agree in a certain statement. The probability that the statement is true is

If from each of the three boxes containing $3$ white and $1$ black, $2$ white and $2$ black, $1$ white and $3$ black balls, one ball is drawn at random. Then the probability that $2$ white and $1$ black balls will be drawn is $\frac{m}{n},$ where $m$ and $n$ are relatively trim then $\left(m+n\right)$ is