Skill: probability-basics
Probability Basics
Probability is a number between 0 and 1 that measures how likely something is — favorable outcomes over total outcomes. Flip coins, roll dice, and spin spinners with confidence.
1 Understand
One formula — what you want over everything possible — answers every theoretical probability question.
What it is
Probability measures how likely an event is, as a number between 0 (impossible) and 1 (certain):
P(event) = favorable outcomes ÷ total possible outcomes.
It works as a fraction, a decimal, or a percent — 1/2 = 0.5 = 50% all say the same thing.
Why it matters
Probability is the language of uncertainty: weather forecasts, game odds, medical risks, and polling margins
all speak it. Every statistics idea after this — sampling, testing, prediction — builds on P(event).
Where it is used
- Games: dice, coins, cards, spinners
- Weather: “40% chance of rain”
- Medicine: risk of side effects
- Polls and election forecasts
2 See It
Two pictures that do the counting for you.
Count what you want over everything
The spinner has 8 equal sections, so every section is equally likely.
3 of them are red, so P(red) = 3/8. The “equal” part matters: if sections had different sizes,
you could not just count them.
Tree diagrams list every path
For two coins, draw the tree: first flip splits into H and T, each splits again.
Four equal paths: HH, HT, TH, TT — each 1/2 × 1/2 = 1/4.
“One heads and one tails” covers two paths (HT and TH), so its probability is 2/4 = 1/2.
3 Worked Examples
Follow each step. The pattern is always the same: count favorable, count total, divide, simplify.
- Favorable faces: 2, 4, 6 — that is 3 out of 6.
- P(even) = 3/6 = 1/2.
Simplify the fraction — 3/6 means the same thing, but 1/2 is the standard answer.
- P(6) = 1/6, and everything must sum to 1.
- P(not 6) = 1 − P(6) = 1 − 1/6 = 5/6.
Direct count agrees: five non-6 faces out of six.
- P(ace) = 4/52 = 1/13. Replacing resets the deck, so P(king) = 1/13 too.
- Independent events multiply: P(both) = 1/13 × 1/13 = 1/169.
Trap check: adding gives 2/13 — but “AND” (both happen) multiplies, “OR” adds.
4 Common Mistakes
Three errors that show up on almost every probability quiz.
“A bag has 1 red and 9 blue marbles, so P(red) = 1/2 — red or blue.”
Count individual equally likely outcomes: P(red) = 1/10.
P(ace then king) = 1/13 + 1/13 = 2/13.
“Both happen” = AND = multiply: 1/13 × 1/13 = 1/169. “One or the other” = OR = add.
P(face card) = 12/52 as the final answer.
12/52 = 3/13 — always reduce to simplest form.
5 Quick Check
Try each one on paper first, then reveal the answer.
One favorable outcome out of two: P(heads) = 1/2 = 50%.
Four paths (HH, HT, TH, TT); only HH works: P = 1/4.
Equivalently, 1/2 × 1/2 = 1/4.
P(red) = 2/5, so P(not red) = 1 − 2/5 = 3/5.
Direct count agrees: 3 blue marbles out of 5.
Key Points to Remember
- P(event) = favorable outcomes ÷ total possible outcomes.
- The formula only works when every outcome is equally likely.
- Complement: P(not E) = 1 − P(E).
- AND multiplies (independent events); OR adds (mutually exclusive events).
- Simplify the probability, just like any fraction.