If D ~ N(250,400) find

(a) P(D>265 ∩ D<280)
(b) P(D>265 | D<280)
(c) P(D<242 ∪ D>256)

I calculated (a) to be 0.0149 and (c) to be 0.986, but I'm not sure about (b)

Thanks for any help

Never mind I figured it out

To find the probabilities, we can first standardize the normal distribution by calculating the z-scores. The z-score is calculated using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.

Given D ~ N(250,400), we can find the z-scores for the given values.

(a) P(D > 265 ∩ D < 280)
To find the probability, we need to find the area between 265 and 280.

Step 1: Calculate the z-scores for 265 and 280.
z1 = (265 - 250) / √400
z2 = (280 - 250) / √400

Step 2: Calculate the probability using the z-scores.
P(D > 265 ∩ D < 280) = P(z1 < Z < z2)

To find this probability, we need to calculate the area under the normal curve between these two z-scores. We can use a standard normal distribution table or a calculator to find this probability.

(b) P(D > 265 | D < 280)
This probability represents the probability of D being greater than 265 given that D is less than 280. In other words, it measures the proportion of the area to the right of 265 among all the area to the left of 280.

P(D > 265 | D < 280) = P(D > 265 ∩ D < 280) / P(D < 280)

We can use the values calculated in the previous steps to find this probability.

(c) P(D < 242 ∪ D > 256)
This probability represents the probability of D being less than 242 or greater than 256. To find this probability, we need to calculate the area to the left of 242 and the area to the right of 256, and then sum these two probabilities.

Step 1: Calculate the z-scores for 242 and 256.
z3 = (242 - 250) / √400
z4 = (256 - 250) / √400

Step 2: Calculate the probabilities using the z-scores.
P(D < 242) = P(Z < z3)
P(D > 256) = P(Z > z4)

P(D < 242 ∪ D > 256) = P(D < 242) + P(D > 256) - P(D < 242 ∩ D > 256)

We can use a standard normal distribution table or a calculator to find these probabilities.

Please note that the values you calculated for (a) and (c) might be approximate, so it's a good idea to double-check them using the steps provided above.