Three urns $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ contains $6$ red and $4$ white; $2$ red and $6$ white; and $1$ red and $5$ white balls respectively. An urn is chosen at random, and a ball is drawn. If the ball drawn is found to be red, find the probability that the balls was drawn from the first urn $\mathrm{A}$.

Three urns contain $2$ white and $3$ black balls; $3$ white and $2$ black balls, and $4$ white and $1$ black ball respectively. One ball is drawn from an urn chosen at random, and it was found to be white. Find the probability that it was drawn from the first urn.

There are three boxes, the first one containing $1$ white, $2$ red and $3$ black balls; the second one containing $2$ white, $3$ red and $1$ black ball and the third one containing $3$ white, $1$ red and $2$ black balls. A box is chosen at random, and from it, two balls are drawn at random. One ball is red and the other, white. What is the probability that they come from the second box?

Urn $\mathrm{A}$ contains $7$ white and $3$ black balls; urn $\mathrm{B}$ contains $4$ white and $6$ black balls; urn $\mathrm{C}$ contains $2$ white and $8$ black balls. One of these urns is chosen at random with probabilities $0.2, 0.6$ and $0.2$ respectively. From the chosen urn, two balls are drawn at random without replacement. Both the balls happen to be white. Find the probability that the balls are drawn are from urn $\mathrm{C}$.

There are $3$ bags, each containing $5$ white and $3$ black balls. Also, there are $2$ bags, each containing $2$ white and $4$ black balls. A white ball is drawn at random. Find the probability that this ball is from a bag of the first group.

There are four boxes, $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$, containing marbles. $\mathrm{A}$ contains $1$ red, $6$ white and $3$ black marbles; $\mathrm{B}$ contains $6$ red, $2$ white and $2$ black marbles; $\mathrm{C}$ contains $8$ red, $1$ white and $1$ black marbles; and $\mathrm{D}$ contains $6$ white and $4$ black marbles. One of the boxes is selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from the box $\mathrm{A}$?

A car manufacturing factory has two plants $\mathrm{X}$ and $\mathrm{Y}$. Plant $\mathrm{X}$ manufactures $70\%$ of the cars, and plant $\mathrm{Y}$ manufactures $30\%$. At plant $\mathrm{X}, 80\%$ of the cars are rated of standard quality, and at plant $\mathrm{Y}, 90\%$ are rated of standard quality.  A car is picked up at random and is found to be of standard quality. Find the probability that it has come from plant $\mathrm{X}$.

An insurance company insured $2000$ scooters and $3000$ motorcycles. The probability of an accident involving a scooter is $0.01$, and that of motorcycles is $0.02$. An insured vehicle met with an accident. Find the probability that the accident vehicle was a motorcycle.

In a bulb factory, machines $\mathrm{A}, \mathrm{B}$ and $C$ manufactures $60\%, 30\%$ and $10\%$ bulbs respectively. Out of these bulbs $1\%, 2\%$ and $3\%$ of the bulbs produced respectively by $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by machine $\mathrm{A}$.