AP Statistics: Binomial and Geometric Random Variables

Binomial Random Variable

A count of successes in a fixed number of trials.

BINS Conditions

Criteria to ensure a random variable is binomial: Binary, Independent, Number, Success.

Binary

Possible outcomes of each trial classified as Success or Failure.

Independent

Trials are independent; the outcome of one trial does not affect another.

Number

The fixed number of trials, denoted as n.

Success

The probability of success (p) is constant across trials.

10% Condition

If the sample size n is less than 10% of the population size N, trials can be treated as independent.

Binomial Probability Formula

P(X = k) = (n choose k) p^k (1 - p)^(n - k).

Binomial Coefficient

Calculated as n!/(k!(n-k)!) and counts arrangements of k successes among n trials.

Mean (Expected Value) - Binomial

μ_X = np, calculates the average of successes.

Standard Deviation - Binomial

σ_X = sqrt(np(1-p)), measures the spread of distribution.

Large Counts Condition

The conditions np >= 10 and n(1-p) >= 10 must be true to approximate a binomial distribution using Normal distribution.

Geometric Distribution

Counts the number of trials until the first success occurs.

Geometric Probability Formula

P(Y = k) = (1 - p)^(k-1)p.

Mean (Expected Value) - Geometric

μ_Y = 1/p, average number of trials until first success.

Standard Deviation - Geometric

σ_Y = sqrt((1 - p) / p), measures spread of distribution.

PDF

Probability Density Function; calculates probability of an exact value.

CDF

Cumulative Distribution Function; accumulates probability from 0 to a value.

TI-84 Function - Binomial Exact

Use binompdf(n, p, k) for P(X = k).

TI-84 Function - Binomial Cumulative

Use binomcdf(n, p, k) for P(X ≤ k).

Calculator Complement Rule

For P(X ≥ k), use 1 - P(X ≤ k - 1).

Common Mistake - At Most/At Least

'At most k' means X ≤ k; 'At least k' means X ≥ k.

Defining the Variable

Always define your variable clearly in Free Response Questions.

Geometric Variable Definition

In geometric distribution, X is the number of trials until first success.

Misidentifying Non-Independent Events

If sampling without replacement, events are not independent; it might be hypergeometric.

Skewness of Binomial Distribution

Symmetric if p = 0.5; skewed right if p < 0.5; skewed left if p > 0.5.

Geometric Distribution Shape

Always skewed right, with the peak probability at the first trial.