Use this probability cheat sheet to choose the right model, substitute values correctly, and check your result. It covers counting, event rules, conditional probability and Bayes’ theorem, random variables, expected value and variance, and the most-used discrete and continuous distributions.
Notation and a reliable solving order
S is the sample space, A and B are events, and Ac is the complement of A. P(A) means the probability of A; P(A|B) means the probability of A given B. For a random variable, X is an outcome, μ is its mean, and σ is its standard deviation.
- Define the random variable or event and its possible outcomes.
- State assumptions such as independence, replacement, a fixed number of trials, or a constant event rate.
- Choose the matching counting rule, event rule, or distribution.
- Substitute values with consistent notation and units.
- Check that probabilities are between 0 and 1, a PMF sums to 1, a PDF integrates to 1, and every conditional denominator is nonzero.
Counting outcomes
Permutations: order matters
For selecting and arranging r objects from n distinct objects:
P(n,r) = n!/(n−r)!
Example: the number of ordered three-letter arrangements from five distinct letters is P(5,3) = 5!/(2!) = 60.
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Combinations: order does not matter
For selecting r objects from n distinct objects without regard to order:
C(n,r) = n!/[r!(n−r)!]
Example: choosing three committee members from five people gives C(5,3) = 10.
Core event-probability rules
Probability axioms
- 0 ≤ P(A) ≤ 1.
- P(S) = 1.
- If A and B are disjoint, P(A ∪ B) = P(A) + P(B).
Complement
P(Ac) = 1 − P(A). This is often the quickest way to calculate “at least one” or “not A.”
Example: If the probability of a defective item is 0.03, the probability that it is not defective is 1 − 0.03 = 0.97.
Addition rule for “A or B”
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Subtract the intersection because outcomes in both events would otherwise be counted twice. If A and B are disjoint, their intersection is zero and the formula reduces to simple addition.
Multiplication rule for “A and B”
P(A ∩ B) = P(A|B)P(B).
The equivalent form P(A ∩ B) = P(B|A)P(A) is useful when the other conditional probability is easier to obtain.
Independence
A and B are independent when learning that one occurred does not change the probability of the other:
P(A ∩ B) = P(A)P(B)
Equivalently, when P(B) > 0, P(A|B) = P(A). Do not assume independence merely because two events are described separately; it must follow from the experiment or be stated.
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Conditional probability and Bayes’ theorem
Conditional probability
When P(B) > 0:
P(A|B) = P(A ∩ B)/P(B).
Example: In a class of 40 students, 12 study physics and 8 both study physics and French. Given that a student studies physics, the probability they also study French is 8/12 = 2/3.
Bayes’ theorem
P(A|B) = P(B|A)P(A)/P(B).
Bayes’ theorem reverses a conditional probability: it updates the probability of a cause A after observing evidence B. The denominator P(B) is the overall probability of the evidence.
Total probability and partition form
If mutually exclusive events A1, A2, …, Ak cover the sample space, then:
P(B) = Σi P(B|Ai)P(Ai)
Therefore:
P(Aj|B) = P(B|Aj)P(Aj) / ΣiP(B|Ai)P(Ai).
Example: A plant receives 60% of parts from supplier 1 and 40% from supplier 2. Defect rates are 1% and 3%, respectively. The overall defect probability is 0.01(0.60) + 0.03(0.40) = 0.018. If a part is defective, the probability it came from supplier 2 is 0.03(0.40)/0.018 ≈ 0.667.
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Discrete random variables
A discrete probability mass function (PMF) assigns a nonnegative probability to each possible value and the probabilities sum to 1:
P(X = x) ≥ 0, and Σ P(X = x) = 1.
Continuous random variables
A continuous probability density function (PDF) is nonnegative and has total area 1:
f(x) ≥ 0, and ∫−∞∞ f(x)dx = 1.
For a continuous variable, probabilities are areas over intervals; the probability of one exact point is zero.
Cumulative distribution function
The CDF gives the probability that X is at most x:
- Discrete: F(x) = Σxi≤x P(X = xi).
- Continuous: F(x) = ∫−∞x f(y)dy.
Expected value, variance, and standard deviation
Expected value (mean)
For a discrete variable:
E[X] = Σ xiP(X = xi)
For a continuous variable:
E[X] = ∫ xf(x)dx
The expected value is the long-term average of repeated observations; it need not be an outcome that can actually occur.
Best Value
Example: A game pays $0 with probability 0.5, $2 with probability 0.3, and $10 with probability 0.2. Its expected payout is 0(0.5) + 2(0.3) + 10(0.2) = $2.60.
Variance
Var(X) = E[(X − E[X])2] = E[X2] − E[X]2.
Variance measures squared spread around the mean. The second form is often faster computationally.
Standard deviation
σ = √Var(X).
Standard deviation is in the same units as X, unlike variance.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Distribution formula table
| Distribution | Use and support | PMF or PDF | Mean | Variance |
|---|---|---|---|---|
| Binomial (n, p) | Number of successes in n independent Bernoulli trials; x = 0,…,n | C(n,x)px(1−p)n−x | np | np(1−p) |
| Hypergeometric | Successes in n draws without replacement from N items, A of them successes | C(A,x)C(N−A,n−x)/C(N,n) | np, where p = A/N | [(N−n)/(N−1)]np(1−p) |
| Geometric (p) | Trial number of the first success; x = 1,2,… | (1−p)x−1p | 1/p | (1−p)/p2 |
| Poisson (μ) | Count of events in a fixed interval with rate μ; x = 0,1,… | e−μμx/x! | μ | μ |
| Uniform (a,b) | Continuous value equally likely on [a,b] | 1/(b−a), for a ≤ x ≤ b | (a+b)/2 | (b−a)2/12 |
| Normal (μ,σ2) | Continuous bell-shaped model; −∞ < x < ∞ | [1/(σ√(2π))]e−(x−μ)2/(2σ2) | μ | σ2 |
| Exponential (rate λ) | Waiting time to an event with constant rate; x ≥ 0 | λe−λx | 1/λ | 1/λ2 |
How to choose a distribution
- Binomial: a fixed number of independent trials, each with the same success probability.
- Hypergeometric: sampling without replacement, so trial outcomes are dependent.
- Geometric: keep repeating identical independent trials until the first success; confirm whether x counts trials or failures.
- Poisson: count events in time, distance, area, or volume when a rate parameter describes random arrivals.
- Uniform: every value in a bounded interval is equally likely.
- Normal: a continuous, unbounded bell-shaped measurement model described by a mean and variance.
- Exponential: a nonnegative waiting time generated by a constant event rate; it is not a bounded-value model.
Before substituting a formula, check six distinctions: discrete versus continuous outcome, support and bounds, sampling with or without replacement, fixed trials versus an event-rate model, independence assumptions, and the mean/variance parameters.
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Quick Recap
Fast error checks
- A probability or PMF entry cannot be negative or exceed 1.
- Mutually exclusive outcomes should add without subtracting an intersection.
- Use the complement for “at least one” when the “none” probability is simpler.
- Do not use binomial formulas for draws without replacement unless an approximation is explicitly justified.
- Do not use an exponential model for a bounded quantity.
- In Bayes’ theorem, identify the base rate P(A) separately from the conditional rate P(B|A).
- For a continuous variable, use interval areas rather than P(X = x).
- Report standard deviation in the original measurement units and variance in squared units.
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