M L Aggarwal Solutions for Chapter: Probability, Exercise 5: EXERCISE

There are three coins. One is a two-headed coin, another is a biased coin that comes up heads $75\%$ of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows heads, then what is the probability that it is the two-headed coin?

Three persons A, B and C apply for a job of Manager role in a Private Company. Chances of their selection (A, B and C) are in the ratio $1: 2: 4$. The probabilities that A, B and C can introduce changes to improve profits of the company are $0.8, 0.5 \mathrm{and} 0.3$ respectively. If the change does not take place, find the probability that it is due to the appointment of C.

By examining the chest X-ray, the probability that TB is detected when a person is actually suffering from it is $0.99$. The probability of an healthy person diagnosed to have TB is $0.001$. In a certain city, $1$ in $1000$ people suffers from TB. A person is selected at random and is diagnosed to have TB. What is the probability that he/she actually has TB?

A doctor is to visit a patient. From the past experience, it is known that the probability that he will come by public transport, scooter, taxi or personal car are respectively $\frac{1}{10}, \frac{1}{5}, \frac{3}{10}$ and $\frac{2}{5}$. The probabilities that he will be late are$\frac{1}{4}, \frac{1}{3}$ and $\frac{1}{12}$ if he comes by public transport, scooter or taxi respectively, but if he comes by personal car he will not be late. When he arrives, he is late. What is the probability that he came by scooter ?

Bag A contains $2$ white, $1$ black and $3$ red balls; Bag B contains $3$ white, $2$ black and $4$ red balls; Bag C contains $4$ white, $3$ black and $2$ red balls. One bag is chosen at random and $2$ balls are drawn at random from that Bag. If the randomly drawn balls are red and black, what is the probability that both balls come from Bag B?

A coin is tossed. If it turns up heads, $2$ balls will be drawn from urn A, otherwise $2$ balls will be drawn from urn B. Urn A contains $3$ black and $5$ white balls. Urn B contains $7$ black balls and $1$ white ball. In both cases, selections are to be made with replacement. What is the probability that urn A is used given that both the balls drawn are black?

A coin is tossed. If it turns up heads, $2$ balls will be drawn from urn A, otherwise $2$ balls will be drawn from urn B. Urn A contains $3$ black and $5$ white balls. Urn B contains $7$ black balls and $1$ white ball. In both cases, selections are to be made with replacement. What is the probability that both the balls drawn are black assuming that the coin is biased and turns up heads $2$ times out of $3$?

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be hearts. Find the probability of the missing card to be a heart.