1. **State the problem:** We have a random variable $x$ that follows a Poisson distribution with parameter $\lambda = 4$. We want to find the probability $P(x=0)$. 2. **Recall the Poisson probability formula:** For a Poisson random variable with parameter $\lambda$, the probability of observing exactly $k$ events is given by: $$P(x=k) = \frac{\lambda^k e^{-\lambda}}{k!}$$ 3. **Apply the formula for $k=0$:** $$P(x=0) = \frac{4^0 e^{-4}}{0!}$$ 4. **Simplify the expression:** Since $4^0 = 1$ and $0! = 1$, we have: $$P(x=0) = e^{-4}$$ 5. **Interpretation:** The probability that $x$ equals zero when $x$ is Poisson distributed with mean 4 is $e^{-4}$. **Final answer:** $$P(x=0) = e^{-4}$$