Binomial Probability Distributions (6.2)

Binomial Distribution

A probability distribution that describes the number of successes in a fixed number of independent trials, with only two outcomes per trial (success/failure) and a constant probability of success.

Characteristics of a Binomial Distribution

1. Only two outcomes per trial (success or failure).
2. Trials are independent.
3. Fixed number of trials n.
4. Constant probability of success p on each trial; probability of failure q = 1 − p.

Binomial Probability Formula

P(X=x)= \binom{n}{x} p^{x} q^{\,n-x} where P(X=x) is the probability of exactly x successes.

n

The total number of trials in a binomial experiment.

p

The probability of a single success on any given trial.

x

The specific number of successes desired or observed (0

\le x

\le n).

q

The probability of failure on any given trial, calculated as q = 1 - p.

Binomial PDF

A calculator function (Binomial PDF(n, p, x)) used to find the probability of exactly x successes in n trials.

When a problem is NOT binomial

If the problem uses a variable total number of trials (e.g., 'until' a certain event occurs with no fixed end), it is not binomial; the number of trials must be fixed in advance.

Unusual Event Cutoff

A probability value (often 0.05) used to define whether an observed event is considered unusual or rare.

Binomial Probability Distribution

A probability distribution characterized by only two outcomes per trial (success/failure), independent trials, a fixed number of trials n, and a constant probability of success p on each trial. The random variable X is the number of successes in n trials.

Binomial Probability Formula

The formula used to calculate the probability of exactly x successes in n trials: P(X=x)= \binom{n}{x} p^{x} q^{\,n-x} where q=1-p.

n (number of trials)

In a binomial distribution, n represents the total number of experiments or trials.

p (probability of success)

In a binomial distribution, p represents the constant probability of a single success on any given trial.

x (number of successes)

In a binomial distribution, x represents the specific number of successes desired or observed, where 0 \le x \le n.

q (probability of failure)

In a binomial distribution, q represents the probability of failure on a single trial, calculated as q = 1 - p.

Binomial PDF (Calculator Function)

A calculator program (e.g., Binomial PDF(n, p, x)) used to compute the probability of exactly x successes in n trials for a binomial distribution.

Conditions for a binomial problem

Key clues include a given percent/probability for a single trial, two outcomes (success/not success), a fixed number of trials that doesn't depend on results, and phrases like 'exactly x', 'not more than', or 'at least'.

When a problem is NOT binomial

A problem is typically not binomial if it involves a variable total number of trials (e.g., 'until' a certain event occurs with no fixed end to the trials).

Unusual Event Cutoff

A probability threshold, often 0.05, below which an event is considered unusual or rare.