Topics 4.3–4.5: Probability, Mutually Exclusive Events & Conditional Probability
Calculating Probabilities, Intersections, and Conditional Events
Video Follow-Along Worksheet
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Learning Objectives:
VAR-4.A: Calculate probabilities for events and their complements.
VAR-4.B: Interpret probabilities for events.
VAR-4.C: Explain why two events are (or are not) mutually exclusive.
VAR-4.D: Calculate conditional probabilities.
Key Vocabulary
Sample Space (S)
The collection of all possible non-overlapping outcomes of a random process
Probability
P(A) = (outcomes in A) / (outcomes in sample space); always between 0 and 1
Complement
The event that A does not happen, denoted A' or Ac; P(A') = 1 - P(A)
Mutually Exclusive
Events that cannot occur at the same time; P(A ∩ B) = 0
Intersection (A ∩ B)
The event that both A and B occur; also called joint probability
Conditional Probability
P(B|A) = probability of B given that A has occurred = P(A ∩ B) / P(A)
Multiplication Rule
P(A ∩ B) = P(A) · P(B|A)
Topic 4.3: Introduction to Probability
[Video: ~9 min]
1.
For a random process, the is the collection of all possible outcomes. It is often denoted by the letter .
2.Sample space examples:
• Flip a single coin: S = {}
• Roll a single die: S = {}
3.
If we flip two coins and record the sequence, our sample space has outcomes: {HH, HT, TH, TT}.
4.
If we roll two dice and record the ordered pair showing on each die, the sample space has possible outcomes.
(6 × 6 = ?)
Probability Formula (Equally Likely Outcomes):
P(A) = number of outcomes in event A total number of outcomes in sample space
5.
A probability is always a number between and , inclusive.
6.
A probability of 0 means the event is . A probability of 1 means it is a .
Listen Up! Record Store Data
An owner of a record store tracks the genre of vinyl albums sold over the past year:
Genre
Rock & Roll
Jazz
Classical
Hip-Hop
World
Pop
Albums Sold
327
431
192
790
276
89
Use this data for questions 7-11.
7.
The random process is randomly selecting an album sold. The sample space is all albums sold, with a total size of albums.
8.
If Event A = "Select a jazz album," then:
P(Jazz) = / ≈
9.Interpreting probability: The probability of an event in a repeatable situation can be interpreted as the with which the event will occur in the .
10.
If we calculate the probability for each genre and add them all up, we should get . This confirms we have a valid .
Complement Rule: The complement of event A (denoted A' or Ac) is the event that A does not happen. P(A') = 1 - P(A)
11.
What is the probability that a randomly selected album is not jazz?
P(not Jazz) = 1 - P(Jazz) = 1 - 0.205 =
Topic 4.4: Mutually Exclusive Events
[Video: ~6 min]
12.
(or ) events cannot occur at the same time.
13.
In the record store example, each album can only be classified in one genre, so the genres are .
Venn Diagrams: Circles represent events within a rectangle (sample space). The overlapping region is the intersection, denoted A ∩ B ("A and B").
14.
The notation A ∩ B represents the of events A and B—outcomes that are in A and B.
15.
If two events are mutually exclusive, there is intersection. The Venn diagram shows two circles that do .
Number Cube Example:
Event A = rolling an odd number: {1, 3, 5}
Event B = rolling a prime number: {2, 3, 5}
Event C = rolling an even number: {2, 4, 6}
16.
P(A ∩ B) = probability of rolling a number that is both odd AND prime = (outcomes: 3 and 5)
17.
P(A ∩ C) = probability of rolling a number that is both odd AND even = . Since P(A ∩ C) = 0, events A and C are .
Super Status! Survey Data
433 high school students answered: "What superpower would you choose?" and "What status would you prefer?"
Superpower
Famous
Happy
Healthy
Rich
Total
Fly
5
48
14
22
89
Freeze Time
3
63
9
26
101
Invisibility
3
62
11
22
98
Super Strength
4
43
17
13
77
Telepathy
0
49
0
19
68
Total
15
265
51
102
433
Use this data for questions 18-22 and 25-30.
18.
P(Happy) = / ≈
19.
P(Happy ∩ Freeze Time) = the probability = / 433 ≈
20.
P(Famous ∩ Telepathy) = / 433 = 0. Since this probability is 0, the events "Famous" and "Telepathy" are .
Key Takeaway: The joint probability is the probability of the intersection of two events. Two events are mutually exclusive if, and only if, P(A ∩ B) = 0.
Topic 4.5: Conditional Probability
[Video: ~8 min]
21.
is the probability that an event happens that another event is known to have already happened.
22.
The notation P(B|A) is read "the probability of given ."
Marble-ous! Example:
A bag contains 10 marbles (4 fully red, 6 not fully red). You select two marbles without replacement.
Event A = first marble is fully red
Event B = second marble is fully red
23.
P(A) = probability first marble is red =
24.
P(B|A) = probability second marble is red, given first was red =
(Only 3 red left out of 9 remaining)
General Multiplication Rule:
P(A ∩ B) = P(A) · P(B|A)
The probability of both A and B = probability of A × probability of B given A
25.
P(both marbles are red) = P(A) · P(B|A) = × =
Conditional Probability Formula:
P(B|A) = P(A ∩ B) / P(A)
This formula can be derived by rearranging the multiplication rule.
26.
Using the Super Status! data, find P(Rich | Fly) — the probability a student wanted to be rich, given they chose Fly.
P(Rich | Fly) = P(Rich ∩ Fly) / P(Fly) = (/433) / (/433) =
27.Shortcut with two-way tables: For P(Rich | Fly), focus only on the "Fly" row. The answer is simply out of .
Order Matters! P(A|B) ≠ P(B|A) in general
28.
P(Fly | Rich) = /
(Focus on the "Rich" column)
29.
P(Fly | Not Rich) requires looking at all columns except Rich.
Students who chose Fly but NOT Rich =
Total students who are NOT Rich =
P(Fly | Not Rich) =
30.
The general multiplication rule P(A ∩ B) = P(A) · P(B|A) can also be written as P(A ∩ B) = P() · P().
Reflection Questions
[AI-Graded]
R1.Using the record store example: If the owner says "About 20% of albums sold are jazz," explain what this probability means in terms of what would happen if we randomly selected many albums. Why is this interpretation more useful than just saying "431 out of 2105"?
R2.Explain the difference between mutually exclusive events and events that simply have a low probability of both occurring. Give an example of each using the Super Status! survey data.
R3.In the marble example, why does the probability of the second marble being red depend on what happened with the first marble? How would this change if we selected marbles WITH replacement instead?
Exit Ticket
Problem: A survey of 200 students asked about their favorite subject (Math or English) and their grade level (Freshman or Sophomore). The results are shown below:
Math
English
Total
Freshman
45
55
100
Sophomore
60
40
100
Total
105
95
200
Calculate P(Math) and interpret what this probability means.
Calculate P(Freshman ∩ Math) — the joint probability.
Are "Freshman" and "Math" mutually exclusive? Explain.
Calculate P(Math | Freshman) and P(Math | Sophomore). What do these tell you?
Class Responses
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