In a normal distribution with a mean of 50 and a standard deviation of 10, calculate the probability of randomly selecting a value between 40 and 60, showing your work.

To calculate the probability of randomly selecting a value between 40 and 60 in a normal distribution with a mean of 50 and a standard deviation of 10, we can use the z-score formula to find the corresponding z-scores for these values and then use a standard normal table (or calculator) to find the probabilities.

The z-score formula is:

z = (X - μ) / σ

where X is the value, μ is the mean, and σ is the standard deviation.

First, let's find the z-scores for 40 and 60:

For X = 40:
z1 = (40 - 50) / 10
z1 = -1

For X = 60:
z2 = (60 - 50) / 10
z2 = 1

Now that we have the z-scores, we can look up their corresponding probabilities in a standard normal table or use a calculator. The probabilities are as follows:

P(z1) = P(Z < -1) ≈ 0.1587
P(z2) = P(Z < 1) ≈ 0.8413

To find the probability of selecting a value between 40 and 60, we need to subtract the probability of z1 from the probability of z2:

P(40 < X < 60) = P(-1 < Z < 1) = P(z2) - P(z1)
P(40 < X < 60) = 0.8413 - 0.1587
P(40 < X < 60) ≈ 0.6826

So, the probability of randomly selecting a value between 40 and 60 in this normal distribution is approximately 68.26%.