The table shows the probability distribution for a discrete random variable $x$, where each $P(x)$ represents the probability associated with each possible value of $x$. The sum of all probabilities in a probability distribution must equal 1.

From the table:

$P(0) + P(1) + P(2) + P(3) = 1$

We know:

$P(1) = 0.25, \quad P(2) = 0.20, \quad P(3) = 0.45$

Let $P(0) = ?$. So, the equation becomes:

$P(0) + 0.25 + 0.20 + 0.45 = 1$

Now, summing the known probabilities:

$0.25 + 0.20 + 0.45 = 0.90$

Thus,

$P(0) + 0.90 = 1$

$P(0) = 1 - 0.90 = 0.10$

So, the missing probability $P(0)$ is 0.10.

Would you like more details or have any questions about this?

Here are 5 related questions:

1. What is a probability distribution?
2. How do you verify that a set of probabilities is valid?
3. What are the properties of a discrete probability distribution?
4. How would this change for a continuous distribution?
5. Can probabilities be negative?

Tip: Always ensure that the sum of all probabilities in any probability distribution equals 1.