Lesson 1: Introduction to Probability
Module 3: Probability Basics • Lesson 1 of 4 • ~30-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define probability and explain what it measures
- Identify sample spaces and events for probability experiments
- Calculate basic probabilities using the classical approach
- Apply fundamental probability rules (0 ≤ P(A) ≤ 1, complement rule)
- Use probability notation correctly
What is Probability?
Probability
Probability is a numerical measure of the likelihood that an event will occur. It quantifies uncertainty with a number between 0 and 1.
Probability helps us answer questions like:
- "What are the chances it will rain tomorrow?" (Weather forecasting)
- "How likely am I to win this game?" (Games and gambling)
- "What's the probability this patient has the disease?" (Medical diagnosis)
- "Will this investment be profitable?" (Finance and risk)
The Probability Scale
- P = 0: Event is impossible (will never happen)
- P = 0.5: Event has a 50-50 chance (equally likely to happen or not)
- P = 1: Event is certain (will definitely happen)
- 0 < P < 1: Event is possible but not certain
Key Terminology
Experiment
An experiment (or trial) is any process that generates well-defined outcomes.
Examples: Flipping a coin, rolling a die, drawing a card, measuring someone's height
Outcome
An outcome is a single result of an experiment.
Examples: Getting "heads" on a coin flip, rolling a "5" on a die
Sample Space (S)
The sample space is the set of all possible outcomes of an experiment.
Examples of Sample Spaces:
| Experiment | Sample Space (S) |
|---|---|
| Flip a coin | S = {Heads, Tails} |
| Roll a six-sided die | S = {1, 2, 3, 4, 5, 6} |
| Draw a card from a standard deck | S = {52 different cards} |
| Count students in a class | S = {0, 1, 2, 3, ...} |
Event
An event is a collection of one or more outcomes from the sample space. We usually denote events with capital letters like A, B, C.
Example: Rolling a Die
Sample Space: S = {1, 2, 3, 4, 5, 6}
Possible Events:
- Event A = "Rolling an even number" = {2, 4, 6}
- Event B = "Rolling a number greater than 4" = {5, 6}
- Event C = "Rolling a 3" = {3}
Calculating Probability: The Classical Approach
Classical Probability Formula
P(A) = (Number of outcomes in event A) / (Total number of outcomes in S)
P(A) = n(A) / n(S)
Where: n(A) = number of outcomes in event A, n(S) = total outcomes in sample space
Important: This formula only works when all outcomes are equally likely!
Examples where it works: Fair coin, fair die, randomly selected card
Examples where it doesn't work: Weighted die, biased coin
Example 1: Rolling a Fair Die
Question: What is the probability of rolling a number greater than 4?
Solution:
- Sample Space: S = {1, 2, 3, 4, 5, 6}, so n(S) = 6
- Event A: "Rolling greater than 4" = {5, 6}, so n(A) = 2
- Calculate: P(A) = n(A) / n(S) = 2 / 6 = 1/3 ≈ 0.333
Answer: The probability is 1/3 or about 33.3%
Example 2: Drawing a Card
Question: What is the probability of drawing a heart from a standard deck?
Solution:
- Sample Space: A standard deck has 52 cards, so n(S) = 52
- Event B: "Drawing a heart" – there are 13 hearts, so n(B) = 13
- Calculate: P(B) = n(B) / n(S) = 13 / 52 = 1/4 = 0.25
Answer: The probability is 1/4 or 25%
Example 3: Flipping Two Coins
Question: What is the probability of getting exactly one head when flipping two coins?
Solution:
- Sample Space: S = {HH, HT, TH, TT}, so n(S) = 4
- Event C: "Exactly one head" = {HT, TH}, so n(C) = 2
- Calculate: P(C) = n(C) / n(S) = 2 / 4 = 1/2 = 0.5
Answer: The probability is 1/2 or 50%
Fundamental Probability Rules
Rule 1: Range of Probabilities
0 ≤ P(A) ≤ 1
Meaning: Every probability must be between 0 and 1 (inclusive).
- If P(A) < 0 or P(A) > 1, you made a mistake!
- Probabilities can be written as decimals (0.25), fractions (1/4), or percentages (25%)
Rule 2: Probability of the Sample Space
P(S) = 1
Meaning: The probability that something in the sample space occurs is 1 (certainty).
Example: When you roll a die, you're guaranteed to get 1, 2, 3, 4, 5, or 6. So P(getting any number) = 1
Rule 3: Probability of an Impossible Event
P(∅) = 0
Meaning: The probability of an impossible event (empty set) is 0.
Example: The probability of rolling a 7 on a standard six-sided die is 0.
Rule 4: Complement Rule
P(not A) = 1 − P(A)
OR: P(A') = 1 − P(A)
Meaning: The probability that event A does NOT occur equals 1 minus the probability that A does occur.
Notation: "not A" can be written as A', Ā, or Ac (complement of A)
Why the Complement Rule is Useful
Sometimes it's easier to calculate P(not A) than P(A)!
Example: "What's the probability of getting at least one head in 3 coin flips?"
- Direct method: Count outcomes with 1, 2, or 3 heads (complicated!)
- Complement method: P(at least one head) = 1 − P(no heads) = 1 − (1/8) = 7/8 (easier!)
Example 4: Using the Complement Rule
Question: If the probability of rain tomorrow is 0.35, what is the probability it will NOT rain?
Solution:
Let A = "it rains tomorrow"
P(A) = 0.35
P(not A) = 1 − P(A) = 1 − 0.35 = 0.65
Answer: The probability it won't rain is 0.65 or 65%
Example 5: Complement with Cards
Question: What is the probability of NOT drawing a face card (Jack, Queen, King) from a standard deck?
Solution:
- There are 12 face cards in a deck (4 suits × 3 face cards)
- P(face card) = 12/52 = 3/13
- P(not a face card) = 1 − P(face card) = 1 − 3/13 = 10/13 ≈ 0.769
Answer: The probability is 10/13 or about 76.9%
Check Your Understanding
Practice Question 1
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If you randomly select one marble, what is the probability of selecting a blue marble?
Solution:
- Total marbles: n(S) = 5 + 3 + 2 = 10
- Blue marbles: n(blue) = 3
- P(blue) = 3/10 = 0.3
Answer: 3/10 or 0.3 or 30%
Practice Question 2
What is the probability of rolling a number less than or equal to 4 on a fair six-sided die?
Solution:
- Sample space: S = {1, 2, 3, 4, 5, 6}, n(S) = 6
- Event A = "rolling ≤ 4" = {1, 2, 3, 4}, n(A) = 4
- P(A) = 4/6 = 2/3 ≈ 0.667
Answer: 2/3 or about 66.7%
Practice Question 3
If P(A) = 0.42, what is P(not A)?
Solution:
Using the complement rule:
P(not A) = 1 − P(A) = 1 − 0.42 = 0.58
Answer: 0.58 or 58%
Practice Question 4
When rolling two six-sided dice, how many outcomes are in the sample space?
Solution:
Each die has 6 possible outcomes.
First die: 6 outcomes
Second die: 6 outcomes
Total outcomes = 6 × 6 = 36
(We'll learn more about this multiplication principle in Lesson 2!)
Answer: 36 outcomes
Practice Question 5
True or False: A probability of 1.25 is possible.
Answer: FALSE
Explanation:
All probabilities must be between 0 and 1 (inclusive).
A probability of 1.25 is greater than 1, which violates the fundamental rule that 0 ≤ P(A) ≤ 1.
If you calculate a probability greater than 1, you've made an error!
Key Takeaways
- Probability measures likelihood with a number between 0 and 1
- Sample space (S) contains all possible outcomes
- Event (A) is a collection of outcomes we're interested in
- Classical formula: P(A) = n(A) / n(S) (when outcomes equally likely)
- Range rule: 0 ≤ P(A) ≤ 1 always
- Complement rule: P(not A) = 1 − P(A)
- Certain event: P(S) = 1
- Impossible event: P(∅) = 0
What's Next?
You've learned the fundamentals of probability! In Lesson 2: Counting Methods, you'll master techniques for systematically counting outcomes—essential for more complex probability problems.
Coming up: Fundamental Counting Principle, Permutations, and Combinations