GATE DA 2025 Set 1 — Question 31
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Probability & Statistics → Probability → Bayes' Theorem
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Question
There are three boxes containing white balls and black balls.Box-1 contains 2 black and 1 white balls.Box-2 contains 1 black and 2 white balls.Box-3 contains 3 black and 3 white balls.In a random experiment, one of these boxes is selected, where the probability of choosing Box-1 is , Box-2 is , and Box-3 is . A ball is drawn at random from the selected box. Given that the ball drawn is white, the probability that it is drawn from Box-2 is ______________ (Round off to two decimal places)
Correct answer
0.25 to 0.25
Solution
Let be the events of selecting Box-1, Box-2, and Box-3 respectively.
Given probabilities:
Let be the event of drawing a white ball.
The probability of drawing a white ball from each box is:
Box-1 (2 Black, 1 White):
Box-2 (1 Black, 2 White):
Box-3 (3 Black, 3 White): We need to find the probability that the ball is drawn from Box-2 given that it is white, i.e., .
Using Bayes' theorem:Substitute the values:
Numerator:
Denominator:
Total Probability
Given probabilities:
Let be the event of drawing a white ball.
The probability of drawing a white ball from each box is:
Box-1 (2 Black, 1 White):
Box-2 (1 Black, 2 White):
Box-3 (3 Black, 3 White): We need to find the probability that the ball is drawn from Box-2 given that it is white, i.e., .
Using Bayes' theorem:Substitute the values:
Numerator:
Denominator:
Total Probability
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