Learning Objectives
- Define probability and explain the 0-to-1 scale
- Calculate probability using the favorable-outcomes formula
- Use the complement rule to find the probability of an event NOT happening
- Build and count sample spaces using organized lists and tables
- Distinguish between theoretical and experimental probability
- Understand why more trials make experimental results converge on theoretical values
What Is Probability?
Probability is the mathematical study of chance — how likely something is to happen. Every time you wonder "what are the odds?", you are thinking about probability.
In everyday language, we say things like "probably," "unlikely," or "no chance." In mathematics, we replace those fuzzy words with precise numbers from 0 to 1.
0 — Impossible
0.25 — Unlikely
0.5 — Even chance
0.75 — Likely
1 — Certain
Examples anchored to the scale:
- Rolling a 7 on a standard 6-sided die: P = 0 (impossible)
- Drawing a red card from a standard deck: P = 0.5 (26 red out of 52)
- The sun rising tomorrow: P = 1 (certain)
- Getting a tail on a fair coin: P = 0.5
- Rolling a 6 on a die: P ≈ 0.167 (1 out of 6)
The Core Formula
Whenever all outcomes are equally likely (a fair coin, an unweighted die, a well-shuffled deck), probability is:
P(event) = Number of favorable outcomes / Total number of possible outcomes
This deceptively simple formula underpins almost everything in basic probability. Let's make it concrete.
Worked Example 1 — Rolling a Die
Problem: What is the probability of rolling an even number on a standard 6-sided die?
Step 1 — List all possible outcomes: {1, 2, 3, 4, 5, 6} → 6 total outcomes
Step 2 — Identify favorable outcomes (even numbers): {2, 4, 6} → 3 favorable outcomes
Step 3 — Apply the formula: P(even) = 3/6 = 1/2 = 0.5
Answer: P(even) = 0.5, or 50%
Worked Example 2 — Drawing a Card
Problem: A standard 52-card deck is shuffled. What is the probability of drawing a King?
Step 1 — Total outcomes: 52 cards
Step 2 — Favorable outcomes: 4 Kings (one per suit: hearts, diamonds, clubs, spades)
Step 3 — P(King) = 4/52 = 1/13 ≈ 0.077
Answer: P(King) ≈ 0.077, or about 7.7% chance
Expressing Probability Three Ways
A probability value can be written as a fraction, a decimal, or a percentage — they all say the same thing.
| Event | Fraction | Decimal | Percentage |
|---|
| Heads on fair coin | 1/2 | 0.5 | 50% |
| Rolling a 3 on a die | 1/6 | 0.167 | 16.7% |
| Drawing a spade | 13/52 = 1/4 | 0.25 | 25% |
| Rolling less than 3 | 2/6 = 1/3 | 0.333 | 33.3% |
| Rolling any number 1-6 | 6/6 = 1 | 1.0 | 100% |
The Complement Rule
The complement of an event is everything that is NOT that event. Since something must either happen or not happen, the two probabilities always add up to exactly 1.
P(not A) = 1 − P(A)
The complement rule is incredibly useful — sometimes it is much easier to calculate what you do NOT want and subtract from 1.
Worked Example 3 — Using the Complement
Problem: A bag contains 3 red marbles and 7 blue marbles. What is the probability of NOT picking a red marble?
Method A (direct): P(blue) = 7/10 = 0.7
Method B (complement): P(red) = 3/10 = 0.3, so P(not red) = 1 − 0.3 = 0.7
Answer: P(not red) = 0.7, or 70% — same result either way
When complement is more useful: What is the probability of rolling at least one 6 in two rolls?
P(at least one 6) = 1 − P(no 6 in either roll) = 1 − (5/6 × 5/6) = 1 − 25/36 = 11/36 ≈ 0.306
Sample Spaces: Listing All Possible Outcomes
A sample space is the complete list of all possible outcomes of an experiment. Building the sample space carefully is essential — miss an outcome and your probability will be wrong.
One Event: Easy to List
For a single fair coin: S = {Heads, Tails} → 2 outcomes.
For a single die: S = {1, 2, 3, 4, 5, 6} → 6 outcomes.
Two Events: Use a Table
When two events happen together (rolling two dice, flipping two coins), listing every combination in a grid is the most reliable method. For two dice, there are 6 × 6 = 36 equally likely outcomes.
Below, the highlighted cells show all outcomes where the two dice sum to 7:
1+1=2
1+2=3
1+3=4
1+4=5
1+5=6
1+6=7
2+1=3
2+2=4
2+3=5
2+4=6
2+5=7
2+6=8
3+1=4
3+2=5
3+3=6
3+4=7
3+5=8
3+6=9
4+1=5
4+2=6
4+3=7
4+4=8
4+5=9
4+6=10
5+1=6
5+2=7
5+3=8
5+4=9
5+5=10
5+6=11
6+1=7
6+2=8
6+3=9
6+4=10
6+5=11
6+6=12
Worked Example 4 — Two Dice Sum
Problem: What is the probability that two fair dice sum to 7?
Step 1 — Total outcomes from the grid: 6 × 6 = 36
Step 2 — Count highlighted outcomes (sum = 7): (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes
Step 3 — P(sum = 7) = 6/36 = 1/6 ≈ 0.167
Answer: About a 16.7% chance. This is actually the most common sum for two dice!
Theoretical vs Experimental Probability
Theoretical probability is what math predicts — calculated from the rules of the experiment, assuming perfect fairness. Experimental probability is what you actually observe by running the experiment.
They are related by this formula:
Experimental P(event) = Number of times event occurred / Total number of trials
Worked Example 5 — Coin Flip Experiment
Suppose you flip a coin 20 times and get heads 9 times.
Theoretical probability of heads = 1/2 = 0.5
Experimental probability of heads = 9/20 = 0.45
The experimental result (0.45) is close to theoretical (0.5) but not exactly equal.
This is normal — random experiments rarely match theory exactly in small samples.
Now flip 1,000 times. You might get 493 heads.
Experimental = 493/1,000 = 0.493 — much closer to 0.5!
Key insight: As the number of trials increases, experimental probability converges toward theoretical probability. This is the Law of Large Numbers.
Why the Difference Exists
In a small number of trials, random variation (also called sampling error) causes results to differ from predictions. This is not a mistake — it is a fundamental feature of probability. Casinos and insurance companies rely on millions of trials, which is why their experimental results converge tightly on theoretical predictions and they make consistent profits.
Certain, Impossible, and Everything Between
Two special probabilities have precise definitions:
- Impossible event: P = 0. Example: rolling a 7 on a standard die. The outcome simply cannot occur.
- Certain event: P = 1. Example: rolling a number between 1 and 6 on a fair die. It must occur.
- Likely event: P > 0.5. The event happens more often than not.
- Unlikely event: P < 0.5. The event happens less often than not.
- Even chance: P = 0.5 exactly. The event happens about half the time.
Common Mistakes to Avoid
- Confusing the formula direction: favorable ÷ total, NOT total ÷ favorable. Order matters.
- Missing outcomes in the sample space: always list systematically, especially for two-event problems.
- Assuming rare events "must happen soon": if heads came up 5 times in a row, the next flip is still 50/50. Past outcomes do not change future probability for independent events (the Gambler's Fallacy).
- Forgetting that probability cannot exceed 1: if your answer is greater than 1, recheck your count of favorable outcomes.
- Treating all outcomes as equal when they are not: a weighted die or a stacked deck changes everything. The formula assumes equal likelihood.
Practice Problems
Problem 1: A bag has 5 green, 3 yellow, and 2 orange marbles. What is P(green)?
Total marbles = 5 + 3 + 2 = 10
Favorable (green) = 5
P(green) = 5/10 = 1/2 = 0.5 = 50%
Problem 2: What is the probability of rolling a number greater than 4 on a single die?
Total outcomes = {1, 2, 3, 4, 5, 6} = 6
Favorable (greater than 4) = {5, 6} = 2
P(greater than 4) = 2/6 = 1/3 ≈ 0.333 ≈ 33.3%
Problem 3: A spinner has 8 equal sections: 3 red, 3 blue, 2 white. What is P(not blue)?
Method 1 (direct): P(not blue) = (3 red + 2 white)/8 = 5/8 = 0.625
Method 2 (complement): P(blue) = 3/8, so P(not blue) = 1 - 3/8 = 5/8 = 0.625
Answer: 5/8 = 62.5%
Problem 4: Two coins are flipped. What is the probability of getting exactly one head?
Sample space: {HH, HT, TH, TT} → 4 equally likely outcomes
Favorable (exactly one head): {HT, TH} = 2 outcomes
P(exactly one head) = 2/4 = 1/2 = 0.5 = 50%
Problem 5: You roll a die 60 times and get a 4 exactly 12 times. What is the experimental probability? How does it compare to theoretical?
Experimental P(4) = 12/60 = 1/5 = 0.2 = 20%
Theoretical P(4) = 1/6 ≈ 0.167 ≈ 16.7%
The experimental value (20%) is a bit higher than theoretical (16.7%). This is normal random variation in 60 trials. With thousands of rolls, the results would converge closer to 16.7%.
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