Wiki Page, Math 13 - Perez Sandbox : Module 12 - Introduction to Normal Probability Distributions (4 of 5)

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2026-01-17T21:15:14-08:00

Now that we have had some practice finding standardized scores, we can use standardized scores to find probabilities.

How do we determine the probability that a value lies some fraction of a standard deviation away from the mean? In this situation, we will use the z-score and technology to find the probability.

We will start with a few examples that walk you through the process of finding a probability using a z-score. Later, we will discuss the nuances of what is actually going on here.

EXAMPLE Length of human pregnancy

The length (in days) of a randomly chosen human pregnancy has a normal probability distribution with , . So X = length of pregnancy (in days).

What is the probability that a randomly chosen pregnancy will last less than 246 days?

The question asks for a probability; therefore, think area.
To find this probability, we first convert X = 246 to a z-score:

The probability that X<246 is equal to the probability that Z<-1.25. This is the area to the left of Z = -1.25. (Why left? Because we are looking for values LESS THAN 246).

Enter Z = -1.25 into the calculator and click Enter. The applet will move the flag to Z = -1.25 and show the corresponding area.

Normal Distribution Calculator

We get an area of 0.1056, which rounds to 0.11. Here is a snapshot of our results with the applet.

Interpretation: There is approximately an 11% chance that a randomly selected pregnancy will last less than 246 days.

EXAMPLE

What is the probability that a pregnancy will last more than 280 days?

The question asks for a probability; therefore, think area.
To find this probability, we first convert X = 280 to a z-score:

Z = (280 – 266) / 16 = 14 / 16 = 0.875

The probability that X > 280 is equal to the probability that Z > 0.875. This is the area to the right of Z = 0.875. (Why to the right? Because we are looking at X-values GREATER THAN 280.)

Enter Z = 0.875 into the calculator and click Enter. The applet will move the flag to Z = 0.875 and show the corresponding area.

Normal Distribution Calculator

We get an area of 0.1908, which rounds to 0.19. Here is a snapshot of our results with the applet.

Interpretation: There is approximately a 19% chance that a randomly selected pregnancy will last more than 280 days.

EXAMPLE

Male foot lengths have a normal probability distribution with a mean of 11 inches and a standard deviation of 1.5 inches.

What is the probability that a randomly selected male has a foot length greater than 13 inches?

The question asks for a probability; therefore, think area.
To find this probability, we first convert X = 13 to a z-score (we've already done this before):

Z = (13 – 11) / 1.5 = 2 / 1.5 = 1.33

The probability that X > 13 is equal to the probability that Z > 1.33. This is the area to the right of Z = 1.33. Enter Z = 1.33 into the calculator and click Enter. The applet will move the flag to Z = 1.33 and show the corresponding area.

Normal Distribution Calculator

We get an area of 0.0918, which rounds to 0.09. Here is a snapshot of our results with the applet.

Interpretation: There is approximately a 9% chance that a randomly selected man will have a foot length greater than 13 inches.

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