To find the probability $P(X < 74)$ for a normally distributed random variable $X$ with mean $\mu = 82$ and standard deviation $\sigma = 4$, we can use the standard normal distribution.

First, we need to convert the value $x = 74$ to a z-score using the formula:

$z = \frac{x - \mu}{\sigma}$

Substituting the given values:

$z = \frac{74 - 82}{4} = \frac{-8}{4} = -2$

Now, we need to find the probability $P(Z < -2)$ where $Z$ is a standard normal variable (mean 0 and standard deviation 1).

Using standard normal distribution tables or a calculator, we find:

$P(Z < -2)$

Looking up the z-score of -2.00 in the standard normal distribution table, we find that the corresponding probability is 0.0228.

Therefore, the probability $P(X < 74)$ is:

$P(X < 74) = 0.0228$

So, the answer is:

$\boxed{0.0228}$

Would you like more details or have any questions?

Here are some related questions to consider:

1. What is the probability that $X$ is greater than 90?
2. Find the probability $P(80 < X < 85)$.
3. Determine the z-score for $X = 70$.
4. Calculate $P(X > 78)$.
5. What is the 95th percentile of this distribution?
6. Find the probability $P(X = 82)$ (Hint: For continuous distributions, the probability at an exact point is 0).
7. If we take a sample of size 30, what is the standard error of the mean?
8. What is the probability that the sample mean of 30 observations is less than 80?

Tip: Always sketch the normal curve and shade the area of interest to visualize the problem better.