The image contains a probability table for a pair of random variables $X$ and $Y$, along with the question to find $P(X = 0 \mid Y = 1)$, the conditional probability.

To solve for $P(X = 0 \mid Y = 1)$, we use the following formula for conditional probability:

$P(X = 0 \mid Y = 1) = \frac{P(X = 0, Y = 1)}{P(Y = 1)}$

Step-by-step approach:

1. Find $P(X = 0, Y = 1)$: From the table, the probability of $X = 0$ and $Y = 1$ is 10/100 (or 0.10).

2. Find $P(Y = 1)$: We sum all probabilities where $Y = 1$. From the table:

   • For $Y = 1$, the probabilities for $X = -1, 0, 1$ are: $\frac{10}{100} + \frac{10}{100} + \frac{8}{100} = \frac{28}{100} = 0.28$

3. Apply the formula: $P(X = 0 \mid Y = 1) = \frac{0.10}{0.28} \approx 0.3571$

So, the conditional probability $P(X = 0 \mid Y = 1)$ is approximately 0.36 (rounded to two decimal places).

Let me know if you need more details or clarification!

Here are some related questions to help further understanding:

1. How do you calculate the joint probability for two variables?
2. What is the difference between marginal and conditional probability?
3. Can you find the probability distribution of $X$ and $Y$?
4. How do you calculate $P(Y = 1)$ using the marginal probability?
5. What would be the result if $P(X = -1 \mid Y = 0)$ were asked?

Tip: When working with conditional probability, remember the formula $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ and make sure the events in the numerator and denominator correspond correctly!