1. Section 3-3
Mutually exclusive events are events that cannot both happen at the same time.
The Addition Rule (For “OR” probabilities)
“Or” can mean one of three things:
A occurs and B does not occur.
B occurs and A does not occur.
Both A and B occur.
If A and B are mutually exclusive, simply add their probabilities of occurring together.
If A and B are NOT mutually exclusive, add their probabilities of occurring together and then subtract the probability that they both occur.

2. Type of Probability & Rules In Words In Symbols
Classical Probability
The number of outcomes in the sample space is known and each outcome is equally likely to occur
𝑃 𝐸 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐸 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
Empirical Probability
The frequency of outcomes in the sample space is estimated from experimentation (you have data)
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝐸 𝑇𝑜𝑡𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝐸 𝑇𝑜𝑡𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑓 𝑛
Range of Probabilities Rule
ALL probabilities are between zero and 1, inclusive
0≤𝑃(𝐸)≤1
Complementary Events
The complement of event E is the set of all outcomes in a sample space that are NOT included in E, denoted E’
𝑃 𝐸′ =1−𝑃 𝐸 ; 𝑃 𝐸 =1−𝑃 𝐸′
Multiplication Rule
The Multiplication Rule is used to find the probability of two events occurring in a sequence (AND)
𝑃 𝐴 𝑎𝑛𝑑 𝐵 =𝑃 𝐴 ∗𝑃 𝐵 𝐴 (𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡)
𝑃 𝐴 𝑎𝑛𝑑 𝐵 =𝑃 𝐴 ∗𝑃 𝐵 (𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡)
Addition Rule
The Addition Rule is used to find the probability of at least one of two events occurring (OR)
𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵 −𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵 (𝑚𝑢𝑡𝑢𝑎𝑙𝑙𝑦 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒)

3. Example 1: Determine whether these two events are mutually
Example 1: Determine whether these two events are mutually exclusive or not.
1) Roll 3 on a die AND roll 4 on a die.
2) Randomly select a male student AND randomly select a nursing major.
3) Randomly select a blood donor with type O blood AND randomly select a female blood donor.
4) Randomly select a jack from a deck of cards AND randomly select a face card from a deck of cards.
5) Randomly select a 20-year-old student AND randomly select a student with blue eyes.
6) Randomly select a vehicle that is a Ford AND randomly select a vehicle that is a Toyota.

4. SOLUTIONS
1) You cannot roll a 3 AND a 4 on the same roll. These are mutually exclusive.
2) You can be both a male and a nursing major. These are not mutually exclusive.
3) You can be both a female and have type 0 blood. These are not mutually exclusive.
4) Jacks are face cards, so any jack is also a face card. These are not mutually exclusive.
5) You can be a 20-year-old student and have blue eyes. These are not mutually exclusive.
6) A vehicle can not be BOTH a Ford AND a Toyota at the same time. These are mutually exclusive.

5. Example 2:
1) You select a card from a standard deck. Find the probability that the card is a 4 or an ace.
2) You select a card from a standard deck. Find the probability that the card is a queen or a red card.
SOLUTIONS
1) “Ace” and “4” are mutually exclusive.
Add the probability of getting an ace to the probability of getting a 4.
The probability of getting an ace is ; the probability of getting a four is also
The probability of getting an ace or a four is
= 8 52 = 2 13 ≈0.154

6. 2) The events are not mutually exclusive (there are 2 red queens).
Add the probability of getting a queen to the probability of getting a red card, then subtract the probability of getting a red queen.
P(queen) = ; P(red) = ; P(red queen) = 2 52
− 2 52 = − 2 52 = ≈0.538

7. Example 3
The frequency distribution shows the volume of sales (in dollars) and the number of months a sales representative reached each sales level during the past three years. If this sales pattern continues, what is the probability that the sales representative will sell between $75,000 and $124,999 next month?
Sales Volume
0-24,999
25,000-49,999
50,000-74,999
75,000-99,999
100, ,999
125, ,999
150, ,999
175, ,999
Months 3 5 6 7 9 2 1

8. Define Event A as monthly sales between $75,000 and $99,999
SOLUTION:
Define Event A as monthly sales between $75,000 and $99,999
Define Event B as monthly sales between $100,000 and $124,999
Because events A and B are mutually exclusive, the probability that the sales representative will sell between $75,000 and $124,999 next month is:
𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵
= = 4 9 ≈0.444
Sales Volume
0-24,999
25,000-49,999
50,000-74,999
75,000-99,999
100, ,999
125, ,999
150, ,999
175, ,999
Months 3 5 6 7 9 2 1

9. Example 4
A blood bank catalogs the types of blood, including positive or negative Rh-factor, given by donors during the last five days. The number of donors who gave each blood type is shown in the table. A donor is selected at random.
1. Find the probability that the donor has type O or type A blood.
2. Find the probability that the donor has type B or is Rh negative.
Blood Type O A B AB
Total
Rh-Factor
Positive
156
139 37 12
344
Negative 28 25 8 4 65
184
164 45 16
409

10. Blood Type O A B AB Total Rh-Factor
SOLUTIONS:
1. These events are mutually exclusive
𝑃 𝑂 𝑜𝑟 𝐴 =𝑃 𝑂 +𝑃 𝐴
= ≈0.851
2. These events are not mutually exclusive.
𝑃 𝐵 𝑜𝑟 𝑅ℎ−𝑛𝑒𝑔 =𝑃 𝐵 +𝑃 𝑅ℎ−𝑛𝑒𝑔 −𝑃(𝐵 𝑎𝑛𝑑 𝑅ℎ−𝑛𝑒𝑔)
− = ≈0.249
Blood Type O A B AB
Total
Rh-Factor
Positive
156
139 37 12
344
Negative 28 25 8 4 65
184
164 45 16
409

11. Example 5
Use the graph to find the probability that a randomly selected draft pick is NOT a running back or a wide receiver.
SOLUTION:
Define Events A and B
Event A: Draft pick is a running back.
Event B: Draft pick is a wide receiver.
These events are mutually exclusive, so the probability that the draft pick is a running back or a wide receiver is:
𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵
= ≈0.231
To find the probability that the draft pick is NOT a running back or wide receiver, simply subtract the probability that he was one of those positions from 1 (complement rule)
1−0.231=0.769

12. Assignment:
Classwork: Pages 165 #1–12 All
Homework: Pages 166 #14–26 Evens