Math Snap

STEP 1

1. $X \sim B(n, p)$ denotes a binomial distribution with parameters $n$ (number of trials) and $p$ (probability of success).
2. The variance of a binomial distribution is given by $\operatorname{Var}(X) = np(1-p)$.
3. The expected value of a binomial distribution is given by $E(X) = np$.
4. We are given that $\operatorname{Var}(X) = 0.8 E(X)$.

STEP 2

1. Express the variance and expected value in terms of $n$ and $p$.
2. Set up the equation based on the given relationship between variance and expected value.
3. Solve the equation for $p$.

STEP 3

Express the variance and expected value in terms of $n$ and $p$:
$$\operatorname{Var}(X) = np(1-p)$$ $$E(X) = np$$

STEP 4

Set up the equation based on the given relationship $\operatorname{Var}(X) = 0.8 E(X)$:
$$np(1-p) = 0.8(np)$$

Solve the equation for $p$:
First, divide both sides by $np$ (assuming $np \neq 0$):
$$1 - p = 0.8$$ Subtract 0.8 from both sides:
$$1 - 0.8 = p$$ $$p = 0.2$$ The value of $p$ is:
$$\boxed{0.2}$$