Explanation
Given Data
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Total number of apples (N): 3 (rotten)+7 (good)=10
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Number of apples drawn (n): 4
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Selection Method: One by one without replacement.
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Random Variable X: The number of rotten apples. Possible values for X are {0,1,2,3}.
1. Probability Distribution
The total number of ways to select 4 apples from 10 is:
10C4=4×3×2×110×9×8×7=210
Now, we calculate the probability for each value of X:
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P(X=0): (0 rotten, 4 good) =2103C0×7C4=2101×35=21035
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P(X=1): (1 rotten, 3 good) =2103C1×7C3=2103×35=210105
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P(X=2): (2 rotten, 2 good) =2103C2×7C2=2103×21=21063
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P(X=3): (3 rotten, 1 good) =2103C3×7C1=2101×7=2107
2. Calculating Mean (μ) and E[X2]
The mean μ (or E[X]) is calculated as ∑xiP(xi):
μ=(0×21035)+(1×210105)+(2×21063)+(3×2107)
μ=2100+105+126+21=210252=1.2
Next, we find E[X2] using ∑xi2P(xi):
E[X2]=(02×21035)+(12×210105)+(22×21063)+(32×2107)
E[X2]=2100+105+252+63=210420=2
3. Final Calculation
The variance σ2 is defined as σ2=E[X2]−μ2.
Rearranging this gives us:
We need to find the value of 10(μ2+σ2):
Correct Option: (4) 20
