Topic 4.6: Independent Events and Unions of Events
Determining Independence & Calculating P(A or B)
Video Follow-Along Worksheet
↻ Already did this one? It stays open even after it’s checked off — just change any answer and click ✓ Check Answers again (no need to Reset). ✨ Grade with AI only ever raises your grade. Make sure you’re signed in so your changes save.
Learning Objective:
VAR-4.E: Calculate probabilities for independent events and for the union of two events.
Key Vocabulary
Independent Events
Events where knowing one occurred does not change the probability of the other
P(A|B) = P(A)
If A and B are independent, the conditional equals the unconditional probability
P(A and B) = P(A) · P(B)
Multiplication rule for independent events only
Union of Events
The probability that event A or event B (or both) will occur: P(A ∪ B)
Addition Rule
P(A or B) = P(A) + P(B) − P(A and B)
Video 1: Determining Independence
[~9 min]
1.
Events A and B are if knowing whether event A has occurred does not change the that event B will occur.
2.
If events are independent, then P(A|B) = and P(B|A) = .
Marble-ous! Example: Select a marble, replace it, then select again. Define A = first marble is red, B = second marble is red. Because we replace, both selections have the same probabilities.
3.
In the marble example with replacement, P(red) = 4/10 = for each draw. The events are independent because replacing keeps probabilities .
Multiplication Rule for Independent Events:
General rule: P(A and B) = P(A) · P(B|A)
For independent events: P(A and B) = P(A) · P(B)
5.
Selecting 4 marbles with replacement: P(red, red, not red, not red) = 0.4 · 0.4 · 0.6 · 0.6 = .
6.
P(at least one red in 10 draws) uses the rule:
= 1 − P(no red in 10) = 1 − (0.6)10 = .
Two Ways to Check Independence:
Check if P(A|B) = P(A) (conditional = unconditional)
Check if P(A and B) = P(A) · P(B) (joint = product)
7.
Given P(E) = 0.40, P(F) = 0.60, P(E and F) = 0.25. Check independence:
Method 1: P(E|F) = P(E and F) / P(F) = 0.25 / 0.60 = .
Since 0.4167 ≠ = P(E), the events are .
8.
Method 2: P(E) · P(F) = 0.4 · 0.6 = .
Since 0.24 ≠ 0.25 = P(E and F), the events are .
Video 2: Unions of Events (Addition Rule)
[~6 min]
9.
The of events A and B is the probability that A or B (or both) will occur, denoted P(A B).
Addition Rule:
P(A or B) = P(A) + P(B) − P(A and B)
10.
Why do we subtract P(A and B)? Because outcomes in events would be counted .
Number Cube Example: Roll a die. A = odd {1, 3, 5}, B = prime {2, 3, 5}. The intersection {3, 5} has 2 outcomes.
11.
P(A or B) = P(odd or prime) = 3/6 + 3/6 − = .
12.
Given P(E) = 0.4, P(F) = 0.6, P(E|F) = 0.25. Find P(E or F).
First find P(E and F) = P(F) · P(E|F) = 0.6 · 0.25 = .
Then P(E or F) = 0.4 + 0.6 − 0.15 = .
13.
From the Super Status table: P(Happy or Super Strength) = (265 + 29 − 15) / 433 = .
14.
If A and B are , then P(A and B) = , so the formula simplifies to P(A or B) = P(A) + P(B).
15.
P(Freeze Time or Super Strength) for mutually exclusive events = 115/433 + 29/433 = .
Video 3: Applying Probability Rules (FRQ Practice)
[~7 min]
What's the News? (2010 AP Exam)
A survey of 2,500 adults asked about education level and primary news source.
College graduates: 693
Get news from internet: 687
College graduates who get news from internet: 245
16.Part A: P(college grad OR internet news)?
Use addition rule: (693 + 687 − 245) / 2500 = .
17.Part B: P(internet | college grad)?
This is probability. Look only at college grads: 245 / 693 = .
18.Key distinction: "An adult is selected" means from the .
"An adult who is a college grad is selected" means from .
19.Part C: Are "college grad" and "internet news" independent?
Compare P(internet | college) = 0.354 to P(internet) = 687/2500 = .
Since they are , the events are .
20.
Alternative check: P(A) · P(B) = (693/2500)(687/2500) = .
P(A and B) = 245/2500 = .
Since 0.076 ≠ 0.098, the events are .
Reflection Questions
[AI-Graded]
R1.Explain the difference between independent events and mutually exclusive events. Can two events be both independent and mutually exclusive? Why or why not?
R2.A student says: "I checked that P(A|B) = P(A), so I know the events are independent." Is checking just one conditional probability enough? Explain what else you might verify.
Exit Ticket
Given the following information about events A and B:
P(A) = 0.3, P(B) = 0.5, P(A and B) = 0.15
Determine: (1) Are A and B independent? Show your work. (2) Find P(A or B).
Class Responses (Esc to close)
Click a "Class" button next to a question to see responses.