Bag I contains 3 Red and 4 Black Balls and Bag II contains 4 Red and 5 Black Balls. One ball is transferred from Bag I to Bag II and then a ball drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
$\frac{16}{31}$
The transferred ball is red with probability \(3/7\) (Bag I has 3 red out of 7) and black with probability \(4/7\).
If the transferred ball is red, Bag II has 5 red and 5 black balls (10 total), so \(P(\text{draw red}\mid\text{transferred red}) = 5/10=1/2\).
If the transferred ball is black, Bag II has 4 red and 6 black balls (10 total), so \(P(\text{draw red}\mid\text{transferred black}) = 4/10=2/5\).
By the law of total probability, \(P(\text{draw red}) = \left(\frac{3}{7}\right)\left(\frac{1}{2}\right)+\left(\frac{4}{7}\right)\left(\frac{2}{5}\right) = \frac{3}{14}+\frac{8}{35} = \frac{15}{70}+\frac{16}{70}=\frac{31}{70}\).
By Bayes' theorem, \(P(\text{transferred black}\mid\text{draw red}) = \dfrac{(4/7)(2/5)}{31/70} = \dfrac{8/35}{31/70} = \dfrac{16}{31}\).
Similar Questions
Probability that A speaks truth is $\frac{4}{5}$ . A coin is tossed. A reports that a head appears. The probability that actually there was head is:
Important Questions from Baye's Theorem
One bag contain 3 white and 2 black balls, another bag contains 5 white and 3 black balls. If a bag is chosen at random and a ball is drawn from it, what is the chance that it is white?
Three cooks X, Y and Z bake a special kind of cake, and with respective probabilities 0.02, 0.03 and 0.05, it fails to rise. In the restaurant where they work, X bakes 50%, Y bakes 30% and Z bakes 20% of cakes. What is the proportion of failures caused by X?
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident involving a scooter driver car driver and a truck driver are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. The probability that the person is a scooter driver is
There are 3 coins in a box. One is a two-headed coin; another is a fair coin; and third is biased coin that comes up heads 75% of time. When one of the three coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?
In a bolt factory, machines X, Y, Z manufacture bolts that are respectively 25% 35%, and 40% of the factory’s total output. The machines X, Y, Z respectively produce 2%, 4% and 5% defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine X?
