Most probability mistakes start before the arithmetic: people add overlapping outcomes or multiply events that cannot occur together. A probability rules cheat sheet helps you choose addition for “or,” multiplication for “and,” complements for “not,” and conditional probability when new information changes the odds.
Choose the rule before you calculate
The right rule depends on the logical words in the question and the relationship between events.
Use this four-question filter
Read the problem once, then ask these questions in order:
- Does it say “not”? Use the Complement Rule: P(not A) = 1 − P(A).
- Does it say “or”? Check whether both events can happen together before adding.
- Does it say “and”? Check whether the events are independent or whether the first changes the second.
- Does it say “given”? Use conditional probability because you are working inside a smaller group.
Fast decision tree:
“Not A” → 1 − P(A).
“A or B” → P(A) + P(B) − P(A ∩ B).
“A and B,” independent → P(A)P(B).
“A and B,” dependent → P(A)P(B|A).
“A given B” → P(A ∩ B) / P(B).
Treat wording as a clue
The Addition Rule of Probability is P(A or B) = P(A) + P(B) − P(A ∩ B). Subtract the intersection because it represents shared outcomes counted twice; if events cannot overlap, the intersection is zero. A complete sample space must total 1 or 100%, which catches impossible answers.
Exclusive and independent are not opposites
Mutually exclusive events cannot happen together, while independent events can happen together without affecting each other.
Compare the event relationship
| Test | Mutually exclusive | Independent |
|---|
| Can both occur? | No | Yes |
| Intersection | 0 | P(A)P(B) |
| “And” result | Always 0 | Multiply |
| Useful wording | One draw, one roll, one slot | Separate random processes |
Work the two quick examples
For one fair die, P(2 or 5) = 1/6 + 1/6 = 1/3 because both results are impossible on one roll. For a fair coin and die, P(heads and 5) = 1/2 × 1/6 = 1/12 because the outcomes are independent.
Add dependent events to the comparison
A probability decision tree should separate three relationships: mutually exclusive, independent, and dependent events. Mutually exclusive events have (P(A /cap B)=0), so they cannot happen on the same trial. Independent events have (P(A /cap B)=P(A)P(B)); language such as “a coin and a die” often signals separate processes. Dependent events occur when the first result changes the second probability, so use (P(A /cap B)=P(A)P(B|A)).
Words such as “without replacement,” “after,” and “remaining” are strong clues. For example, drawing two aces from a deck without replacement is ((4/52)(3/51)), not ((4/52)(4/52)). Dependent events may still overlap, but their intersection must account for the changed probability of the second event.
Conditional probability needs the base rate
Conditional probability asks how likely A is after B is known: P(A|B) = P(A ∩ B) / P(B), where P(B) must exceed zero.
If a condition affects 1% of people, a test detects 90% of cases, and 5% of unaffected people test positive, then 10,000 people produce about 90 true positives and 495 false positives. A positive result therefore means roughly a 15% chance of the condition, not 90%.
Use evidence without magical thinking
Bayes’ Theorem updates a base rate after evidence: P(A|B) = [P(B|A) × P(A)] / P(B). It does not promise an individual outcome; check whether evidence truly changes the odds before treating a streak, setback, or rare win as meaningful.
This sheet is not enough for confidence intervals, regression, complex dependence among many variables, or probabilities that cannot be estimated credibly. A population probability also cannot guarantee what will happen in one person, one investment, or one game.
Total probability, distributions, and expected value
Use the Law of Total Probability when one outcome can arise through several non-overlapping paths. If the groups (A_1, A_2, /dots) cover the whole sample space, then (P(B)=/sum P(B|A_i)P(A_i)). For example, if 40% of orders ship from Warehouse A with a 2% damage rate and 60% ship from Warehouse B with a 5% damage rate, (P(/text{damage})=(0.02)(0.40)+(0.05)(0.60)=0.038), or 3.8%. This combines conditional probabilities into an overall probability.
A discrete probability distribution lists every possible value and its probability; its probabilities must total 1. Its expected value is (E(X)=/sum xP(X=x)), the long-run average. For a fair die, (E(X)=3.5), even though 3.5 is not a possible roll.
Common questions
What are the basic rules of probability?
Each probability is from 0 to 1, a complete sample space totals 1, and P(not A) = 1 − P(A). Use addition for “or,” multiplication for “and,” and conditional probability after a stated condition.
What are the five rules of probability?
Five core rules are the range rule, sample-space rule, Addition Rule, Multiplication Rule, and Complement Rule. Conditional probability is also essential because it changes the reference group.
What is the addition rule in probability?
The addition rule is P(A or B) = P(A) + P(B) − P(A ∩ B). Subtract the intersection only if both events can occur together.
What is the multiplication rule in probability?
For independent events, P(A and B) = P(A)P(B). For dependent events, use P(A and B) = P(A)P(B|A).
What is the difference between exclusive and independent events?
Mutually exclusive events cannot occur together, so their intersection equals 0. Independent events can occur together, and one leaves the other’s probability unchanged.
How do I calculate conditional probability?
Calculate it with P(A|B) = P(A ∩ B) / P(B), as long as P(B) exceeds 0. Work only within cases where B occurred.
Can I print this cheat sheet on one page?
Yes, print in portrait mode at 85% to 90% scale, or use your browser’s Print command and choose “Save as PDF.”
Use probability to make cleaner choices
The essentials:- Check overlap before adding probabilities, or shared outcomes get counted twice.
- Multiply only after deciding whether events are independent or dependent.
- Read “given” as a signal to narrow the group and inspect the base rate.
- Reject answers below 0, above 1, or based on impossible intersections.
Quick self-check
1. A card is drawn from a standard deck. What is P(heart or king)?
There are 13 hearts and 4 kings, but the king of hearts is counted in both groups. Use the Addition Rule: /(13/52+4/52-1/52=16/52=4/13/).2. A bag has 3 red and 2 blue marbles. What is P(two red without replacement)?
The first draw changes the composition of the bag, so these are dependent events: /(P(R /cap R)=(3/5)(2/4)=3/10/).3. If P(rain) = 0.30, what is P(no rain)?
Apply the complement rule: /(1-0.30=0.70/). A valid answer must remain between 0 and 1.
Learn more
Here are some additional resources on this subject:
