Mean and Variance of the Number of Good Eggs

To find the mean and variance of the number of good eggs, we need to consider the probability distribution of the number of good eggs drawn without replacement from a total of 8 eggs, where 2 are rotten.

Probability Distribution
Let's define the random variable X as the number of good eggs drawn without replacement. X can take values from 0 to 3, as there are only 3 draws without replacement.

To calculate the probabilities for each value of X, we can use the hypergeometric probability distribution formula:

P(X = x) = (C(g, x) * C(r, n - x)) / C(N, n)

Where:
- N is the total number of eggs (8)
- n is the number of draws without replacement (3)
- g is the number of good eggs (8 - 2 = 6)
- r is the number of rotten eggs (2)
- C(a, b) represents the combination of a things taken b at a time

Calculating Mean
The mean of a random variable represents the average value and is given by the formula:

Mean (μ) = Σ (x * P(X = x))

We can calculate the mean by multiplying each value of X with its corresponding probability and summing them up.

Mean (μ) = (0 * P(X = 0)) + (1 * P(X = 1)) + (2 * P(X = 2)) + (3 * P(X = 3))

Calculating Variance
The variance of a random variable measures the spread of the distribution and is given by the formula:

Variance (σ²) = Σ ((x - μ)² * P(X = x))

We can calculate the variance by subtracting the mean from each value of X, squaring the result, multiplying by its corresponding probability, and summing them up.

Variance (σ²) = ((0 - μ)² * P(X = 0)) + ((1 - μ)² * P(X = 1)) + ((2 - μ)² * P(X = 2)) + ((3 - μ)² * P(X = 3))

Calculations
Let's calculate the mean and variance using the formulas and the given values:

P(X = 0) = (C(6, 0) * C(2, 3 - 0)) / C(8, 3) = (1 * 2) / 56 = 1/28
P(X = 1) = (C(6, 1) * C(2, 3 - 1)) / C(8, 3) = (6 * 2) / 56 = 6/28 = 3/14
P(X = 2) = (C(6, 2) * C(2, 3 - 2)) / C(8, 3) = (15 * 2) / 56 = 30/56 = 15/28
P(X = 3) = (C(6, 3) * C(2, 3 - 3)) / C(8, 3) = (20 * 1) / 56 = 20/

Kuch or saval nahi tha kya