Question

Three bad eggs got mixed up with 7 good eggs. If three eggs are drawn (without replacement) from 10 eggs, find the mean and variance for the number of bad eggs among them.

Solution

Number of good eggs = 7
Number of bad eggs = 3
∴ total number of eggs = 7 + 3 = 10
Let X denote the random variable ‘bad egg is obtained’. X takes the values 0, 1, 2, 3.
$straight P left parenthesis straight X space equals space 0 right parenthesis space equals space 7 over 10 cross times 6 over 9 cross times 5 over 8 space equals 210 over 720 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space 3 over 10 cross times 7 over 9 cross times 6 over 8 plus 7 over 10 cross times 3 over 9 cross times 6 over 8 plus 7 over 10 cross times 6 over 9 cross times 3 over 8 space equals 126 over 720 plus 126 over 720 plus 126 over 720 space equals space 378 over 720 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space 3 over 10 cross times 2 over 9 cross times 7 over 8 plus 3 over 10 cross times 7 over 9 cross times 2 over 8 plus 7 over 10 cross times 3 over 9 cross times 2 over 8 space equals 42 over 720 plus 42 over 720 plus 42 over 720 space equals 126 over 720 straight P left parenthesis straight X space equals space 3 right parenthesis space equals space 3 over 10 cross times 2 over 9 cross times 1 over 8 space equals 6 over 720$
$therefore$ X takes the values 0, 1, 2, 3 with probabilities $210 over 720 comma space 378 over 720 comma space 126 over 720 comma space 6 over 720$

$straight mu space equals space sum from blank to blank of straight x space straight p subscript straight i space equals space 648 over 720 space equals space 9 over 10 straight sigma squared space equals space sum from blank to blank of straight x squared space straight p subscript straight i space minus space straight mu squared space equals space 936 over 720 minus 81 over 720 space equals space 13 over 10 minus 81 over 100 space equals space fraction numerator 130 minus 81 over denominator 100 end fraction space equals space 49 over 100 therefore space space space space space mean space equals space straight mu space equals space 9 over 10 comma space space variance space equals space 49 over 100$

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Probability

Three bad eggs got mixed up with 7 good eggs. If three eggs are drawn (without replacement) from 10 eggs, find the mean and variance for the number of bad eggs among them.

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