Why Learn Probability?
Probability is the mathematics of uncertainty — and uncertainty is everywhere. Weather forecasts ("40% chance of rain"), medical tests ("95% accurate"), insurance premiums, game strategy, and scientific research all depend on probabilistic reasoning. Even everyday decisions involve probability intuitively. This lesson gives you the precise vocabulary and calculation methods to move from gut feeling to mathematical certainty about uncertain events.
By the end of this lesson you will be able to write formal probabilities, construct sample spaces, identify complementary and compound events, and compare theoretical predictions to experimental results.
Step-by-Step Lesson
Every probability is a number between 0 and 1 (or equivalently, 0% to 100%).
- P = 0: The event cannot happen (rolling a 7 on a standard die).
- P = 1: The event is certain (rolling a number between 1 and 6).
- P = 0.5: Equal chance of happening or not (fair coin landing heads).
Probabilities are expressed as fractions, decimals, or percentages — but the underlying value is always between 0 and 1 inclusive.
The sample space (S) is the set of all possible outcomes of an experiment. The event (E) is the set of outcomes we are interested in.
Worked Example — Standard Die
A fair six-sided die. Sample space S = {1, 2, 3, 4, 5, 6}.
Event: rolling an even number → E = {2, 4, 6} → 3 favorable outcomes.
P(even) = 3/6 = 1/2 = 0.5 = 50%
Event: rolling a number greater than 4 → E = {5, 6} → 2 favorable.
P(>4) = 2/6 = 1/3 ≈ 0.333 ≈ 33.3%
For equally likely outcomes, listing the sample space is the most reliable way to count correctly. Never skip this step for unfamiliar problems.
The complement of event A (written A' or "not A") consists of all outcomes NOT in A. Since every outcome is either in A or not in A:
Worked Example — Drawing a Card
A standard 52-card deck. What is the probability of NOT drawing a heart?
P(heart) = 13/52 = 1/4
P(not heart) = 1 − 1/4 = 3/4 = 0.75 = 75%
Complementary thinking is powerful: when direct calculation is complex, find the complement instead. "What is the probability of rolling at least one 6 in two rolls?" is easier as 1 − P(no 6 in two rolls).
Compound events combine two or more events. For independent events (one does not affect the other), the rules are:
Worked Example — Two Coins
Flip two fair coins. P(heads on first) = 1/2. P(heads on second) = 1/2.
P(both heads) = 1/2 × 1/2 = 1/4
Sample space: {HH, HT, TH, TT} — confirming 1 of 4 outcomes is HH.
First flip Second flip Outcome
H ────────── H HH ← P = 1/4
│ T HT ← P = 1/4
T ────────── H TH ← P = 1/4
T TT ← P = 1/4
P(at least one head) = P(HH) + P(HT) + P(TH) = 3/4. Or: 1 − P(TT) = 1 − 1/4 = 3/4. Same answer, complement is faster.
Experimental (empirical) probability is calculated from actual observations:
Worked Example
A student flips a coin 50 times and gets heads 27 times.
Experimental P(heads) = 27/50 = 0.54 = 54%
Theoretical P(heads) = 1/2 = 0.50 = 50%
The gap (4%) is normal for 50 trials. By 10,000 flips the experimental result will be very close to 50% — this is the Law of Large Numbers: as trials increase, experimental probability converges to theoretical probability.
When two dice are rolled, there are 6 × 6 = 36 equally likely outcomes. Listing all 36 in a grid is the safest method for complex questions.
The grid below shows all outcomes when summing two dice. Highlighted cells (green) show outcomes where the sum = 7:
P(sum = 7) = 6/36 = 1/6 ≈ 16.7%. Seven is the most probable sum on two dice — important to know for board games and probability problems alike.
Practice Problems
Problem 1
A bag contains 5 red, 3 blue, and 2 green marbles. What is the probability of drawing a red marble? A non-red marble?
Total marbles = 5 + 3 + 2 = 10
P(red) = 5/10 = 1/2 = 0.5
P(not red) = 1 − 1/2 = 1/2 = 0.5
Or directly: P(not red) = (3 + 2)/10 = 5/10 = 1/2. Both methods agree.
Problem 2
A spinner has 8 equal sections numbered 1–8. What is the probability of spinning a prime number?
Sample space: {1, 2, 3, 4, 5, 6, 7, 8}
Prime numbers in range: {2, 3, 5, 7} — 4 primes (1 is not prime)
P(prime) = 4/8 = 1/2 = 0.5
Problem 3
You roll a fair die twice. What is the probability of getting a 3 on the first roll AND a 3 on the second roll?
The two rolls are independent.
P(3 on first) = 1/6
P(3 on second) = 1/6
P(both 3) = 1/6 × 1/6 = 1/36 ≈ 2.78%
Problem 4
In an experiment, a thumbtack is dropped 200 times and lands point up 130 times. What is the experimental probability of landing point up? Point down?
P(point up) = 130/200 = 0.65 = 65%
P(point down) = 1 − 0.65 = 0.35 = 35%
Or: (200 − 130)/200 = 70/200 = 0.35. Note: for a thumbtack, there is no "theoretically correct" probability — experimental data is the only source of truth.
Problem 5
What is the probability of rolling two dice and getting a sum greater than 9?
Total outcomes = 36
Sums > 9: 10, 11, 12
Sum = 10: (4,6),(5,5),(6,4) = 3 ways
Sum = 11: (5,6),(6,5) = 2 ways
Sum = 12: (6,6) = 1 way
Total favorable = 3 + 2 + 1 = 6
P(sum > 9) = 6/36 = 1/6 ≈ 16.7%
5 Common Mistakes
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1Adding probabilities for AND events
P(A and B) = P(A) × P(B) for independent events, NOT P(A) + P(B). Addition is for OR events. Mixing these rules is the most common probability error.
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2Thinking probability tells you what WILL happen
P(heads) = 0.5 does not mean every other flip is heads. Probability describes long-run patterns, not the outcome of any single trial. You can get 10 heads in a row with a fair coin.
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3Incomplete sample space
Missing possible outcomes leads to probabilities that don't add to 1. Always verify: the sum of all outcome probabilities equals exactly 1.
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4Not replacing the item when calculating successive draws
If you draw a card and don't replace it, the second draw's sample space has 51 cards, not 52. "With replacement" and "without replacement" give different answers.
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5Assuming equally likely outcomes when they are not
A thumbtack, a loaded die, or a weighted spinner does NOT have equally likely outcomes. The theoretical formula P = favorable/total only applies when all outcomes are equally likely.