Explanation
1. Problem Setup
We have three identical boxes:
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Box 1 (B1): Contains 2 Golden balls {G,G}.
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Box 2 (B2): Contains 2 Silver balls {S,S}.
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Box 3 (B3): Contains 1 Golden ball and 1 Silver ball {G,S}.
Since the boxes are identical, the probability of choosing any box is:
2. Individual Conditional Probabilities
The probability of drawing a Golden ball (G) from a specific box:
3. Total Probability of Drawing a Golden Ball
First, we calculate the total probability of selecting a golden ball, P(G):
P(G)=P(B1)P(G∣B1)+P(B2)P(G∣B2)+P(B3)P(G∣B3)
P(G)=(31×1)+(31×0)+(31×21)
4. Final Answer (Posterior Probabilities)
Using Bayes' Theorem, we find the probability that the golden ball came from a specific box:
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For Box 1:
P(B1∣G)=P(G)P(B1)P(G∣B1)=1/21/3×1=32
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For Box 2:
P(B2∣G)=P(G)P(B2)P(G∣B2)=1/21/3×0=0
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For Box 3:
P(B3∣G)=P(G)P(B3)P(G∣B3)=1/21/3×1/2=31
The conditional probabilities are 2/3, 0, and 1/3 respectively.
