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A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected(with replacement) at random.The probability that the first sample contains exactly 1 black marble,and the second sample contains exactly 3 black marbles is?
Verified Answer
A bag contains a large number of white and black marbles in equal port...
Probability of black = 1/2
probability of white = 1/2
probability = 
= 5/32*5/16
= 25/512
This question is part of UPSC exam. View all JEE courses
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A bag contains a large number of white and black marbles in equal port...
Problem:
A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected (with replacement) at random. Find the probability that the first sample contains exactly 1 black marble, and the second sample contains exactly 3 black marbles.
Solution:
To solve this problem, we will break it down into two separate events - the first sample containing exactly 1 black marble, and the second sample containing exactly 3 black marbles. We will calculate the probability of each event and then multiply them together to find the overall probability.
Event 1: First sample contains exactly 1 black marble
Let's assume that there are x black marbles and x white marbles in the bag. The total number of marbles in the bag is 2x.
To choose exactly 1 black marble in the first sample, we need to choose 1 black marble and 4 white marbles. The probability of choosing 1 black marble is given by:
P(1 black marble) = (x/2x) * (x/2x) * (x/2x) * (x/2x) * ((2x-x)/2x) = (x^4 * (x-x))/(2x)^5 = x^4/(2x)^5 = 1/(2x)
Event 2: Second sample contains exactly 3 black marbles
Since we are selecting with replacement, the probability of selecting a black marble remains the same for each selection. The probability of choosing exactly 3 black marbles in the second sample is given by:
P(3 black marbles) = (x/2x) * (x/2x) * (x/2x) * ((2x-x)/2x) * ((2x-x)/2x) = (x^3 * (2x-x) * (2x-x))/(2x)^5 = x^3/(2x)^5 = 1/(2x)^2
Overall probability:
To find the overall probability, we need to multiply the probability of event 1 and event 2:
P(1 black marble and 3 black marbles) = P(1 black marble) * P(3 black marbles) = (1/(2x)) * (1/(2x)^2) = 1/(2x)^3
Therefore, the probability that the first sample contains exactly 1 black marble and the second sample contains exactly 3 black marbles is 1/(2x)^3.
A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected (with replacement) at random. Find the probability that the first sample contains exactly 1 black marble, and the second sample contains exactly 3 black marbles.
Solution:
To solve this problem, we will break it down into two separate events - the first sample containing exactly 1 black marble, and the second sample containing exactly 3 black marbles. We will calculate the probability of each event and then multiply them together to find the overall probability.
Event 1: First sample contains exactly 1 black marble
Let's assume that there are x black marbles and x white marbles in the bag. The total number of marbles in the bag is 2x.
To choose exactly 1 black marble in the first sample, we need to choose 1 black marble and 4 white marbles. The probability of choosing 1 black marble is given by:
P(1 black marble) = (x/2x) * (x/2x) * (x/2x) * (x/2x) * ((2x-x)/2x) = (x^4 * (x-x))/(2x)^5 = x^4/(2x)^5 = 1/(2x)
Event 2: Second sample contains exactly 3 black marbles
Since we are selecting with replacement, the probability of selecting a black marble remains the same for each selection. The probability of choosing exactly 3 black marbles in the second sample is given by:
P(3 black marbles) = (x/2x) * (x/2x) * (x/2x) * ((2x-x)/2x) * ((2x-x)/2x) = (x^3 * (2x-x) * (2x-x))/(2x)^5 = x^3/(2x)^5 = 1/(2x)^2
Overall probability:
To find the overall probability, we need to multiply the probability of event 1 and event 2:
P(1 black marble and 3 black marbles) = P(1 black marble) * P(3 black marbles) = (1/(2x)) * (1/(2x)^2) = 1/(2x)^3
Therefore, the probability that the first sample contains exactly 1 black marble and the second sample contains exactly 3 black marbles is 1/(2x)^3.
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A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected(with replacement) at random.The probability that the first sample contains exactly 1 black marble,and the second sample contains exactly 3 black marbles is?
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A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected(with replacement) at random.The probability that the first sample contains exactly 1 black marble,and the second sample contains exactly 3 black marbles is? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected(with replacement) at random.The probability that the first sample contains exactly 1 black marble,and the second sample contains exactly 3 black marbles is? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected(with replacement) at random.The probability that the first sample contains exactly 1 black marble,and the second sample contains exactly 3 black marbles is?.
A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected(with replacement) at random.The probability that the first sample contains exactly 1 black marble,and the second sample contains exactly 3 black marbles is? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected(with replacement) at random.The probability that the first sample contains exactly 1 black marble,and the second sample contains exactly 3 black marbles is? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A bag contains a large number of white and black marbles in equal portions. Two samples of 5 marbles are selected(with replacement) at random.The probability that the first sample contains exactly 1 black marble,and the second sample contains exactly 3 black marbles is?.
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