Theorem of Total Probability

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.

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.

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 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:

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 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 $\mathrm{COVID}-19$ ?

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

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 with strike, then find the probability that construction job will be completed on time.

Given $n\left(U\right)=25, n\left(A\right)=14, n\left(B\right)=10$ and $(A\cap B)=5$, where $U$ is the universal set, $A$ and $B$ are subsets of $U$. Then what will be the $n\left(\overline{A\cup B}\right)$?

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

An urn containing '14' green and '6' pink ball. K (< 14, 6) balls are drawn and laid a side, their colour being ignored. Then one more ball is drawn. Let P(E) be the probability that it is a green ball, then 10 P(E) =..............

An urn contains 5 red and 2 green balls, one ball is chosen from urn, if it is red then a green ball is put back into Box, and if it is green then a red ball is put in to box (previous ball was not put in the box), now a second ball is drawn from the urn. The probability that it is red ball is

A box contains $b$ blue balls and  $r$ red balls. A ball is drawn randomly from the box and is returned to the box with another ball of the same colour. The probability that the second ball drawn from the box is blue, is

Two persons $A$ and $B$ throw a (fair) die $($six-faced cube with faces numbered from $1$ to $6$$)$ alternately, starting with $A$. The first person to get an outcome different from the previous one by the opponent wins.The probability that $B$ wins is,

Urn $A$ contains $6$ Red and $4$ Black balls and urn $B$ contains $4$ Red and $6$ Black balls. One ball is drawn at random from urn $A$ and placed in urn $B$. Then one ball is drawn at random from urn $B$ and placed in urn $A$. If one ball is now drawn at random from urn $A$, then the probability that it is Red is

A candidate takes three tests in succession and the probability of passing the first test is $p.$ The probability of passing each succeeding test is $p$ or $\frac{p}{2}$ according as he passes or fails in the preceding one. The candidate is selected, if he passes atleast two tests. The probability that the candidate is selected, is

Urn $A$ contains $6$ red and $4$ black balls and urn $B$ contains $4$ red and $6$ black balls. One ball is drawn at random from $A$ and placed in $B.$ Then one ball is drawn at random from $B$ and placed in $A.$ If one ball is now drawn from A then the probability that it is found to be red is

One bag contains 5 white and 3 black balls and an other contains 4 white, 5 red balls. Two balls are drawn from one of them choosing at random. The probability that they are of different colours is