Home / Courses / Algebra 2 / Unit 9

Algebra 2 Unit 9: Probability & statistics

This unit covers counting outcomes, computing probabilities and using the normal distribution.

Covers: Counting principle, permutations and combinations, probability rules, normal distribution, data collection

Study guide

The fundamental counting principle says that if one choice can be made m ways and another n ways, together they can be made m·n ways. A permutation counts arrangements where order matters: P(n, r) = n!/(n − r)!. A combination counts selections where order doesn't matter: C(n, r) = n!/(r!(n − r)!). Ask yourself whether rearranging the same items gives a different outcome; if so, use permutations.

Probability is favorable outcomes divided by total equally likely outcomes, between 0 and 1. The complement rule says P(not A) = 1 − P(A), which makes "at least one" problems easy: 1 − P(none). The addition rule is P(A or B) = P(A) + P(B) − P(A and B). For independent events, P(A and B) = P(A)·P(B); for dependent events, multiply by the conditional probability, P(B given A). Two-way tables help organize conditional probabilities.

The normal distribution is a symmetric, bell-shaped curve described by its mean and standard deviation. The empirical rule says about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. A z-score, (x − mean)/SD, tells how many standard deviations a value is from the mean.

Data can come from surveys, observational studies or experiments. Random sampling avoids bias and lets you generalize; random assignment in an experiment lets you conclude cause and effect.

A parameter is a number that describes a whole population, such as the true mean of all voters. A statistic is a number calculated from a sample and used to estimate the parameter. For boundaries that aren't whole standard deviations from the mean, use a calculator's normal cumulative distribution function with the lower bound, upper bound, mean and standard deviation. Multiply the resulting proportion by the population size to estimate a count.

In a simulation, the middle 95% of simulated sample results (roughly the mean ± 2 standard deviations) gives a range of plausible values. If an observed result falls inside this interval, it is consistent with the claim; if it falls outside, the result is unusual and casts doubt on the claim. Events A and B are independent if P(A and B) = P(A)·P(B), or equivalently if P(A | B) = P(A). With a two-way table, compare the conditional probability with the overall probability; if they are equal (or very close), the events are independent.

Key ideas

• Counting principle: multiply the number of choices at each step.
• Permutations when order matters; combinations when it doesn't.
• P(not A) = 1 − P(A). "At least one" is usually easiest with the complement.
• Empirical rule: 68-95-99.7 within 1, 2, 3 standard deviations.
• Random samples allow generalization; random assignment allows causation.

Formulas and rules

Common mistakes

• Using permutations when order doesn't matter.
• Adding probabilities of events that overlap.

Words to know

Combinations formula

C(n, r) = n!/(r!(n − r)!)

Permutation

An arrangement where order matters: ₙPᵣ = n!/(n − r)!.

Empirical rule

About 68%, 95%, 99.7% within 1, 2, 3 SDs.

P(at least one)

1 − P(none)

Factorial

n! = n·(n − 1)···2·1 (5! = 120).

Fundamental counting principle

m ways then n ways gives m·n ways.

Combination

A selection where order doesn't matter: C(n, r) = n!/(r!(n − r)!).

Independent events

P(A and B) = P(A)·P(B).

Addition rule

P(A or B) = P(A) + P(B) − P(A and B).

Normal distribution

A symmetric bell-shaped curve set by its mean and standard deviation.

Z-score

(x − mean)/SD: standard deviations from the mean.

Random sample

Every member has an equal chance of selection; avoids bias.

Complement rule

P(not A) = 1 − P(A); use it for "at least one."

Data collection methods

Surveys, observational studies and experiments.

Sample practice questions

33 more questions like these, from level 1 to 5, are in the practice set.