Probability Notes – Normal Probabilities
Review the difference between inequality symbols less than (points to left), less
than or equal to, greater than (points to right), greater than or equal to.
Discrete Probability – Probabilities that involve countable totals. Think categorical
variables like winning and losing. For example the probability of rolling a die and
getting a three. This we can count. There is 1 side that has three dots out of 6
total sides, so the probability is 1/6
Continuous Probability – Probabilities about numbers in a continuous scale.
Quantitative data is in involved that falls in a continuous scale. For example the
probability that a person’s height is greater than 1.602719 meters. Notice there
are infinitely many possibilities for this decimal. So we cannot directly count the
total. The total is infinite. Also the probability that someone’s height is exactly
1.602719 meters is about zero.
Continuous probabilities are difficult to calculate. We use a probability density
curve.
Probability Density Curve: A curve of a particular shape where the area under the
curve is equal to 1 (or 100%). So to find probabilities we can either use calculus or
technology to calculate the area under the curve.
Normal Probability Density Curve: For continuous quantitative data that is bell
shaped, we can use the normal calculator in StatCrunch. Remember this
corresponds to probabilities for continuous quantitative data when it is
impossible to count the total.
What do we Need?
Bell shaped quantitative data
Mean and Standard Deviation (remember only accurate when bell shaped)
The number you are finding the probability of is X
Notation:
P ( X ) means probability of X happening. For example, find the probability that a
person is more than 1.602719 meters tall. P ( X > 1.602719 ). Remember the
probability of equals is about zero in continuous data so P ( X > 1.602719 ) is
about the same as P ( X ≥ 1.602719 ).
Normal Probabilities with StatCrunch:
 Go to Stat button and click on “Calculator”.
 Then go down the menu to “Normal”.
 Enter the mean and standard deviation for your bell shaped data.
 Decide on “standard” ( for probabilities involving ≤ or ≥ ) or click “between”
(for probabilities in between two x values).
 Put in either the x value or the probability given. If you put the x value,
StatCrunch will calculate the probability (Area). If you put the probability
(Area), StatCrunch will calculate the x value. Put whether you want greater
than or equal to or less than or equal to. Remember probability of equal is
zero.
Example 1: Suppose heights of men are bell shaped with a mean average of 1.76
meters with a standard deviation of 0.071 meters. Draw a bell shaped curve with
1.76 in the center. Find the probability that the height of a man is greater than or
equal to 1.602719 meters? (Mark off this area on your curve)
P ( X ≥ 1.602719 ) =
The curve should look like the one you drew. The answer is 0.987 or 98.7% of
men.
Example 2: Now find the height in meters that about 15% of men are less than.
Plug in 0.15 where the probability goes and make sure to press the less than or
equal to.
P ( X ≤ ?? ) = 0.15
You should get a height of about 1.686 meters.
Example 3: Now find the probability that a man is between 1.8 meters and 1.9
meters tall. Remember to click the between button and use the same mean and
standard deviation. Draw the bell shaped curve with 1.76 in middle. Where is 1.8
and 1.9. Estimate. Shade the area under the curve.
Now put 1.8 and 1.9 into the between normal calculator and get the answer.
P ( 1.8 ≤ X ≤ 1.9 ) =