Probability Rules
Master the addition rule, multiplication rule, complementary events, and mutually exclusive events.
Fundamental Probability Rules
Probability measures how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain). The following rules help us calculate probabilities for combined events.
The Core Rules
Complementary Rule
P(A') = 1 - P(A)
The probability that event A does not happen equals 1 minus the probability it does.
Addition Rule (General)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
The probability of A or B occurring. Subtract the overlap to avoid double-counting.
Multiplication Rule (Independent Events)
P(A ∩ B) = P(A) × P(B)
If A and B are independent, the probability of both occurring is the product.
Mutually Exclusive Events
Two events are mutually exclusive if they cannot occur at the same time. When rolling a single die, getting a 3 and getting a 5 are mutually exclusive because you cannot roll both simultaneously.
Mutually Exclusive vs Non-Mutually Exclusive
Mutually Exclusive
No overlap: P(A ∩ B) = 0
P(A ∪ B) = P(A) + P(B)
Not Mutually Exclusive
Overlap exists: P(A ∩ B) > 0
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Complementary Events in Practice
The complement of event A (written A' or A̅) contains all outcomes not in A. This is especially useful when calculating "at least one" probabilities, where finding the complement is much simpler.
Strategy: "At Least One"
P(at least one) = 1 - P(none)
For example, the probability of rolling at least one six in 4 rolls = 1 - P(no sixes in 4 rolls) = 1 - (5/6)4.
Key Vocabulary
Mutually Exclusive
Events that cannot both occur at the same time. P(A ∩ B) = 0.
Complement
The event A' contains all outcomes not in A. P(A') = 1 - P(A).
Independent Events
Events where the occurrence of one does not affect the probability of the other.
Union (A ∪ B)
The event that A or B (or both) occurs. Calculated using the addition rule.
Worked Examples
P(A) = 0.4, P(B) = 0.3, and A and B are mutually exclusive. Find P(A ∪ B).
Step 1: Since A and B are mutually exclusive, P(A ∩ B) = 0.
Step 2: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.3 - 0 = 0.7
Answer: P(A ∪ B) = 0.7
A coin is tossed 3 times. What is the probability of getting at least one head?
Step 1: Use the complement: P(at least 1 head) = 1 - P(no heads).
Step 2: P(no heads) = P(all tails) = (1/2)3 = 1/8
Step 3: P(at least 1 head) = 1 - 1/8 = 7/8
Answer: P(at least one head) = 7/8 = 0.875
P(A) = 0.5, P(B) = 0.6, P(A ∩ B) = 0.2. Find P(A ∪ B).
Step 1: Use the addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Step 2: = 0.5 + 0.6 - 0.2 = 0.9
Answer: P(A ∪ B) = 0.9
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
If P(A) = 0.7, what is P(A')?
Question 2
A die is rolled. Events: A = {even number}, B = {number > 4}. Are A and B mutually exclusive?
Question 3
Two independent events have P(A) = 0.3 and P(B) = 0.5. Find P(A ∩ B).
Question 4
P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.1. What is P(A ∪ B)?
Question 5
A bag contains 3 red and 7 blue marbles. Two marbles are drawn with replacement. What is the probability both are red?
Key Concepts Summary
- ●The complementary rule: P(A') = 1 - P(A).
- ●The addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
- ●For mutually exclusive events: P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
- ●For independent events: P(A ∩ B) = P(A) × P(B).
- ●Use complements for "at least one" problems: P(at least one) = 1 - P(none).