Bayes' Theorem

A bag contains $6$ balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least $5$ black balls is

The urns $A, B$ and $C$ contains $4$ red, $6$ black; $5$ red, $5$ black and $\lambda$ red, $4$ black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn $C$ is $0.4$, then the square of length of the side of largest equilateral triangle, inscribed in the parabola $y^2=\lambda x$ with one vertex at vertex of parabola is

$25\%$ of the population are smokers. A smoker has $27$ times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$.Then the value of $k$ is _____ .

Bag $\mathrm{I}$ contains $3$ red, $4$ black and $3$ white balls and Bag $\mathrm{II}$ contains $2$ red, $5$ black and $2$ white balls. One ball is transferred from Bag $\mathrm{I}$ to Bag $\mathrm{II}$ and then a ball is draw from Bag $\mathrm{II}$. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is

Bag $A$ contains $2$ white, $1$ black and $3$ red balls and bag $B$ contains $3$ black, $2$ red and $n$ white balls. One bag is chosen at random and $2$ balls drawn from it at random are found to be $1$ red and $1$ black. If the probability that both balls come from Bag $A$ is $\frac{6}{11}$, then $n$ is equal to _____

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is $0.9$ and that of the second unit is $0.8.$ The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $p,$ then $98p$ is equal to

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

In a group of $400$ people, $160$ are smokers and non-vegetarian; $100$ are smokers and vegetarian and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35\%,20\%$ and $10\%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

Out of 11 consecutive natural number if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference is :

Box $1$ contains $30$ cards numbered $1$ to $30$ and Box $2$ contains $20$ cards numbered $31$ to $50$. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box $1$ is 

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered $\mathrm{1,} 2, 3,\dots , 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is

A box $A$ contains $2$ white, $3$ red and $2$ black balls. Another box $B$ contains $4$ white, $2$ red and $3$ black balls. If two balls are drawn at random, without replacement from a randomly, selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box $B$ 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: