Understanding the Binomial Distribution
The Binomial Distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. It is defined by two primary parameters:
- $n$: The number of trials.
- $p$: The probability of success in a single trial.
The Fundamental Formulas
For any Binomial Distribution $X \sim B(n, p)$:
- Mean (Expected Value): $\mu = np$
- Variance: $\sigma^2 = np(1-p)$
Step-by-Step Problem Solution
Given the problem: Find the binomial distribution having mean = 12 and variance = 8.
Step 1: Set up the equations
From the definitions above, we have:
- $np = 12$
- $np(1-p) = 8$
Step 2: Solve for $p$
We can substitute $np$ from the first equation into the second: $(12)(1-p) = 8$ $1-p = \frac{8}{12}$ $1-p = \frac{2}{3}$ $p = 1 - \frac{2}{3} = \frac{1}{3}$
Step 3: Solve for $n$
Now, substitute $p = 1/3$ back into the mean equation ($np = 12$): $n(\frac{1}{3}) = 12$ $n = 12 \times 3 = 36$
Conclusion
The Binomial Distribution is $B(36, 1/3)$. This means there are 36 trials with a success probability of 1/3.