To compute the probability $P(0 \leq z \leq 0.81)$ where $z$ is a standard normal random variable, we can use the standard normal distribution table (also known as the Z-table) or a statistical calculator.

Steps:

1. Find $P(z \leq 0.81)$:
   Use the Z-table or technology to find the cumulative probability for $z = 0.81$.

   From the Z-table, $P(z \leq 0.81) \approx 0.7910$.

2. Find $P(z \leq 0)$:
   The cumulative probability for $z = 0$ is always $0.5$, since the standard normal distribution is symmetric around zero.

3. Calculate $P(0 \leq z \leq 0.81)$:
   Subtract the probability of $P(z \leq 0)$ from $P(z \leq 0.81)$: $P(0 \leq z \leq 0.81) = P(z \leq 0.81) - P(z \leq 0) = 0.7910 - 0.5000 = 0.2910$

Final Answer:

$P(0 \leq z \leq 0.81) = 0.2910$

Would you like any more details or explanations on this?

Here are some related questions you might find useful:

1. How do we use the Z-table to find probabilities for other values of $z$?
2. What is the relationship between the standard normal distribution and other normal distributions?
3. How can technology (e.g., graphing calculators or software) simplify finding these probabilities?
4. How do we handle probabilities when $z$-values are negative?
5. What are the common uses of the standard normal distribution in real-world scenarios?

Tip: The Z-table gives cumulative probabilities from the far left up to the value of $z$, so for intervals, you'll need to subtract probabilities like we did here!