Probability Lesson—INSTRUCTOR SOLUTIONS
MATH 123 Instructor Training
Probabilities have rules. Discuss these rules as they relate to the table you created above:
1. Probabilities are always between 0 and 1
a. What does a 0 probability represent?
b. What does a probability of 1 mean?
2. The sum of the probabilities of all unique possibilities is 100%
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒊𝒕𝒆𝒎𝒔 𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕
3. Probabilities can be written as ratios/fractions:
𝒕𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒑𝒐𝒔𝒔𝒊𝒃𝒊𝒍𝒊𝒕𝒊𝒆𝒔
Consider the following data on police officers:
Table 2
Promoted
in the last year
Not Promoted
in the last year
Total
Male
30
90
120
Female
20
60
80
50
150
200
Total
Fill in the missing quantities.
Circle each group of numbers that sum to 200.
What is the probability a police officer selected at random (think names in a hat) is female?
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑡𝑒𝑚𝑠 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
=
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠
80
200
= 0.40
Answer the following using the data from Table 2 above:
120
1. What is the probability a police officer selected at random is male? 200 = 0.60
2. What is the probability a police officer selected at random was promoted in the last year?
50
= 0.25
200
150
3. What is the probability a police officer was not promoted in the last year? 200 = 0.75
4. How many police officers represented here are women who were promoted? Notice you are
looking for officers who are women and were promoted. 20
Now find the probability of selecting at random a woman who was promoted in the last year?
20
200
= 0.10
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Probability Lesson—INSTRUCTOR SOLUTIONS
MATH 123 Instructor Training
5. How many police officers are men who were promoted in the last year? Again you are looking
for officers who are men and were promoted. 30
Now find the probability of selecting at random a man who was promoted in the last year?
30
200
= 0.15
6. What is the probability of selecting a man who was not promoted in the last year? How could
you restate this question with the word “and” in it?
90
200
= 0.45 Officers who are male AND not
promoted in the last year.
7. Locate and circle the area in the table which represent the and quantities:
Male
Female
Total
Promoted Not Promoted
30
90
20
60
Total
200
Using NOT when calculating probabilities: Consider an allergy test which has an accuracy rate of 90%.
What is the chance the test is inaccurate? __10%_. Write a rule for calculating NOT probabilities:
___________________Probability of NOT X = 100% - probability of X_____________________
8. What is the probability of selecting an officer who was not female? 1-P(female) = 1-0.40=0.60
9. What is the probability of selecting a police officer who was not not promoted? 1-P(not
promoted)=1-0.75=0.25
10. What is the likelihood of randomly selecting a police officer who is not a promoted man? 1P(promoted man)=1-0.15=0.85
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Probability Lesson—INSTRUCTOR SOLUTIONS
MATH 123 Instructor Training
Conditional Probabilities: Consider a condition a change in the denominator. If a police officer is a
man, what is the probability he was promoted? The condition is that he is a man. The ratio becomes:
𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛
𝒑𝒓𝒐𝒎𝒐𝒕𝒆𝒅 𝒎𝒆𝒏
𝒎𝒆𝒏
=
𝟑𝟎
𝟏𝟐𝟎
= 𝟎. 𝟐𝟓
20
11. If a police officer is a woman, what is the probability she was promoted in the last year? 80 =
0.25
12. If a police officer is a woman, what is the probability she was not promoted in the last year?
60
80
= 0.75
20
13. If a police officer was promoted in the last year, what is the probability it was a woman? 50 =
0.40
14. If a police officer was promoted in the last year, what is the probability it was a man?
30
50
= 0.60
90
15. If an officer was not promoted in the last year, what is the probability it was a man?150 = 0.60
Using OR when calculating probabilities: Consider circling everything that would be included and check
to see if there is a double count.
16. What is the probability of randomly selecting a police officer who is female or was not promoted?
Promoted Not Promoted
30
90
20
60
Total
Promoted Not Promoted
30
90
20
60
Total
Male
Female
Total
200
17. What is the probability of selecting an officer who is either male or female?
Male
Female
Total
150 + 80 − 60
= 0.85
200
120 + 80
= 1.0
200
200
18. What is the probability of randomly choosing a police officer who is either promoted or male?
Male
Female
Total
Promoted Not Promoted
30
90
20
60
Total
200
3
50 + 120 − 30
= 0.70
200
Probability Lesson—INSTRUCTOR SOLUTIONS
MATH 123 Instructor Training
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