Probability Checklist

Probability is the language connecting quantum states to measurement outcomes. Quantum mechanics changes how probabilities are computed, but it does not remove the need to understand normalization, expectation values, variance, conditioning, and probability densities.

This checklist focuses on the classical probability skills needed before reading the Born rule, expectation values, density matrices, and measurement pages.

You Should Be Able To

• Normalize a finite probability distribution.
• Normalize a probability density.
• Distinguish probabilities from probability densities.
• Compute expectation values for discrete and continuous variables.
• Compute variances and standard deviations.
• Use conditional probability.
• Apply Bayes’ rule in simple cases.
• Recognize independent and correlated variables.
• Interpret a covariance.
• Distinguish an ensemble distribution from a single outcome.
• Check that probabilities are nonnegative and sum or integrate to one.

Why This Matters in Quantum Mechanics

The Born rule assigns probabilities to measurement outcomes. Expectation values are averages over possible outcomes, not necessarily values obtained in a single trial. Variance measures the spread of a prepared state’s outcome distribution. Density operators describe mixtures and subsystems. Open systems, quantum information, and statistical mechanics all rely on probabilistic reasoning.

The recurring mistake is to treat a probability amplitude as a probability. Amplitudes can add and interfere; probabilities are nonnegative real numbers obtained after the appropriate quantum rule is applied.

Minimum Examples

You should be comfortable with:

Task	Example
Normalize probabilities	$p_1+p_2+p_3=1$
Normalize a density	$\int_a^b \rho(x)\,dx=1$
Compute an expectation	$\mathbb E[X]=\sum_x x\,p(x)$
Compute a variance	$\operatorname{Var}(X)=\mathbb E[X^2]-\mathbb E[X]^2$
Use conditional probability	$P(A\mid B)=P(A\cap B)/P(B)$
Distinguish density from probability	$P(a<X<b)=\int_a^b \rho(x)\,dx$

Diagnostic Problems

1. Let a discrete random variable take values $-1,0,1$ with probabilities $1/4,1/2,1/4$. Compute its expectation value and variance.

Solution

The expectation value is

$$\mathbb E[X] =(-1)\frac14+0\frac12+1\frac14 =0.$$

The second moment is $\mathbb E[X^2]=1/4+1/4=1/2$, so

$$\operatorname{Var}(X) =\mathbb E[X^2]-\mathbb E[X]^2 =\frac12.$$

2. Let $\rho(x)=2x$ on $0<x<1$. Verify normalization and compute $\mathbb E[X]$.

Solution

Normalization:

$$\int_0^1 2x\,dx=1.$$

The expectation value is

$$\mathbb E[X] =\int_0^1 x(2x)\,dx =2\int_0^1 x^2\,dx =\frac{2}{3}.$$

3. A test detects a condition with probability $0.9$ if the condition is present and has false-positive probability $0.1$ if it is absent. If the prior probability of the condition is $0.2$, what is $P(\text{condition}\mid\text{positive})$?

Solution

Bayes’ rule gives

$$P(C\mid +) =\frac{P(+\mid C)P(C)} {P(+\mid C)P(C)+P(+\mid \neg C)P(\neg C)} =\frac{0.9\cdot0.2}{0.9\cdot0.2+0.1\cdot0.8} =\frac{0.18}{0.26} =\frac{9}{13}.$$

4. Why is $\rho(x)$ not itself the probability of finding a particle at exactly $x$?

Solution

For a continuous variable, probabilities are assigned to intervals. The probability of finding $X$ in $[a,b]$ is $\int_a^b\rho(x)\,dx$. The density $\rho(x)$ can have units and can be larger than one; it becomes a probability only after integration over a region.

Where to Review

Use these pages when a checklist item is weak:

• Probability Spaces: Light Version
• Random Variables
• Probability Densities
• Expectation Values
• Variance and Covariance
• Conditional Probability
• Bayes’ Rule
• Classical vs Quantum Probability
• Born Rule

References

• W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, 1968.
• S. Ross, A First Course in Probability, 10th ed., Pearson, 2018.
• A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed., McGraw-Hill, 2002.