The image shows a normal distribution curve with three different probability shading tasks. I'll analyze each one as shown:

1. First Task: $P(\mu - 3\sigma < x < \mu - 2\sigma)$

   • This represents the probability of a random variable $x$ lying between $\mu - 3\sigma$ and $\mu - 2\sigma$.
   • In the graph, the shaded region is to the left, and it covers the interval from $\mu - 3\sigma$ to $\mu - 2\sigma$, indicating a small portion of the left tail.

2. Second Task: $P(x \geq \mu)$

   • This represents the probability of $x$ being greater than or equal to the mean $\mu$.
   • The graph should ideally show shading from $\mu$ to the right, covering the entire right half of the normal distribution (for values greater than or equal to $\mu$). However, the image appears to have shading on the left, which might be incorrect for this probability.

3. Third Task: $P(x < \mu + \sigma)$

   • This represents the probability that $x$ is less than $\mu + \sigma$.
   • The shaded area should cover everything to the left of $\mu + \sigma$ (extending from the far left tail up to just before $\mu + \sigma$).

Would you like more detailed solutions for each probability or further clarifications on standard deviation and normal distribution properties?

Here are some related questions for further exploration:

1. How do we calculate the probability for intervals in a normal distribution?
2. What does it mean if a value falls within one, two, or three standard deviations of the mean?
3. How can we interpret a right-tailed or left-tailed probability in a normal distribution?
4. Why is the area under the normal distribution curve equal to 1?
5. How does changing the mean or standard deviation affect the normal distribution curve?

Tip: For a standard normal distribution (mean 0, standard deviation 1), you can use Z-scores to calculate probabilities.