Question

Mean + Variance = 1.8, n = 5, Find p(probability of success).

Answer 1

1. Understand the Problem:
We're given that the mean plus the variance of a binomial distribution is 1.8. We're also given that n (the number of trials) is 5. We need to find p (the probability of success).

2. Recall Binomial Distribution Formulas:
Mean (μ): μ = np
Variance (σ²): σ² = np(1 - p)

3. Set Up the Equation:
We are given: Mean + Variance = 1.8
So, np + np(1 - p) = 1.8

4. Substitute the Given Value of n:
We know n = 5, so substitute that into the equation:
\[ 5p + 5p(1 - p) = 1.8 \]

5. Simplify and Solve for p:
\[ 5p + 5p - 5p² = 1.8 \] \[ 10p - 5p² = 1.8 \] \[ 5p² - 10p + 1.8 = 0 \] Multiply by 10 to eliminate the decimal: \[ 50p² - 100p + 18 = 0 \] Divide by 2 to simplify: \[ 25p² - 50p + 9 = 0 \]

6. Use the Quadratic Formula to Solve for p:
The quadratic formula is: \[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In our equation, a = 25, b = -50, and c = 9.
\[ p = \frac{50 \pm \sqrt{(-50)^2 - 4 \cdot 25 \cdot 9}}{2 \cdot 25} \] \[ p = \frac{50 \pm \sqrt{2500 - 900}}{50} \] \[ p = \frac{50 \pm \sqrt{1600}}{50} \] \[ p = \frac{50 \pm 40}{50} \]

7. Find the Two Possible Values of p:
\[ p_1 = \frac{50 + 40}{50} = \frac{90}{50} = 1.8 \quad (\text{This is not possible, as probability must be between 0 and 1}) \] \[ p_2 = \frac{50 - 40}{50} = \frac{10}{50} = \frac{1}{5} = 0.2 \]

8. Check Validity:
Since probability must be between 0 and 1, p = 0.2 is the only valid solution.

Therefore, the probability of success (p) is 0.2.