It looks like you've uploaded a math problem involving uniform distributions and probability. Based on the image you shared, the question is:

Suppose $X$ is uniformly distributed over $[0, 5]$ and $Y$ is uniformly distributed over $[0, 4]$. If $X$ and $Y$ are independent, then $P(\max(X, Y) > 3)$ is...

Let's solve this step by step:

1. The probability of $X$ and $Y$ being independent means that we can multiply their individual probabilities.
2. Since $X \sim \text{Uniform}(0, 5)$ and $Y \sim \text{Uniform}(0, 4)$, we know their probability density functions (PDFs):
   • The PDF for $X$, $f_X(x) = \frac{1}{5}$ for $x \in [0, 5]$.
   • The PDF for $Y$, $f_Y(y) = \frac{1}{4}$ for $y \in [0, 4]$.

Now, the problem asks for the probability that the maximum of $X$ and $Y$ is greater than 3, i.e., $P(\max(X, Y) > 3)$.

This is the complement of the event where both $X$ and $Y$ are less than or equal to 3. Hence: $P(\max(X, Y) > 3) = 1 - P(X \leq 3, Y \leq 3)$

Now, we calculate $P(X \leq 3, Y \leq 3)$ under the assumption that $X$ and $Y$ are independent: $P(X \leq 3, Y \leq 3) = P(X \leq 3) \cdot P(Y \leq 3)$

For $X \sim \text{Uniform}(0, 5)$, $P(X \leq 3) = \frac{3}{5}$.

For $Y \sim \text{Uniform}(0, 4)$, $P(Y \leq 3) = \frac{3}{4}$.

Thus: $P(X \leq 3, Y \leq 3) = \frac{3}{5} \times \frac{3}{4} = \frac{9}{20}$

Finally, the probability we need is: $P(\max(X, Y) > 3) = 1 - \frac{9}{20} = \frac{11}{20}$

So the probability is $\frac{11}{20}$.

Let me know if you'd like further clarification!

Here are 5 related questions for further exploration:

1. How can you calculate probabilities when random variables are dependent?
2. What happens to this probability if the distributions of $X$ and $Y$ change?
3. How do you calculate the cumulative distribution function (CDF) for uniform distributions?
4. How would you approach a similar problem if $X$ and $Y$ were normally distributed?
5. What is the relationship between the mean of a uniform distribution and the range of values?

Tip: When calculating probabilities involving the maximum or minimum of two independent random variables, it's often helpful to use the complement rule, as shown in this solution!