Section 6.2: The Binomial Probability Distribution – Study Notes

Section 6.2: The Binomial Probability Distribution

Objective 1: Determine Whether a Probability Experiment is a Binomial Experiment

The binomial probability distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. Binomial experiments are foundational in statistics for modeling binary outcomes.

• Criteria for a Binomial Experiment:

  1. The experiment consists of a fixed number of trials, n.

  2. Each trial is independent.

  3. Each trial has only two possible outcomes: success or failure.

  4. The probability of success, p, is the same for each trial.

• Key Terms:

  • n: Number of trials

  • p: Probability of success on a single trial

  • q: Probability of failure on a single trial ($q = 1 - p$)

• Binomial Random Variable: If X is the number of successes in n trials, possible values of X are 0, 1, ..., n.

• Example:

  • A basketball player makes 80% of free throws. In 3 attempts, the number of successful free throws is a binomial random variable.

  • Survey: 25% of Americans prefer a certain flavor of ice cream. In a sample of 10, the number who prefer that flavor is binomial.

  • Drawing cards without replacement is not binomial (trials are not independent).

Objective 2: Use the Binomial Probability Distribution Function

The binomial probability distribution function calculates the probability of obtaining exactly x successes in n independent trials.

• Binomial Probability Formula:

  • $P(X = x) = \binom{n}{x} p^x q^{n-x}$

  • Where $\binom{n}{x}$ is the binomial coefficient: $\binom{n}{x} = \frac{n!}{x!(n-x)!}$

  • n: Number of trials

  • x: Number of successes

  • p: Probability of success

  • q: Probability of failure ($q = 1 - p$)

• Math Symbols:

  Phrase	Math Symbol
  Greater than or equal to	≥
  Less than or equal to	≤
  Greater than	>
  Less than	<
  Exactly or equals	=

• Example:

  • Given 45% of smartphone owners use a feature, in a sample of 15, calculate the probability that exactly 12 use it, fewer than 8 use it, or at least 9 use it using the binomial formula.

Objective 3: Compute the Mean and Standard Deviation of a Binomial Random Variable

The mean and standard deviation of a binomial random variable provide measures of central tendency and spread for the distribution of successes.

• Formulas:

  • Mean (Expected Value): $\mu = np$

  • Standard Deviation: $\sigma = \sqrt{npq}$

• Example:

  • In a sample of 50 smartphone owners, with 45% using a feature, the mean and standard deviation of the number who use it are calculated using the above formulas.

Objective 4: Graph a Binomial Probability Distribution

Graphing a binomial probability distribution involves plotting the probability of each possible number of successes. The shape of the distribution depends on the values of n and p.

• Steps to Graph:

  1. Calculate $P(X = x)$ for each possible value of x (from 0 to n).

  2. Plot the probabilities on a bar graph.

• Shape of Distribution:

  • If $p < 0.5$, the distribution is skewed right.

  • If $p = 0.5$, the distribution is symmetric.

  • If $p > 0.5$, the distribution is skewed left.

• Example:

  • Graphing with $n = 10$, $p = 0.2$ (skewed right), $p = 0.5$ (symmetric), $p = 0.8$ (skewed left).

• Distribution Shape and Number of Trials:

  • As n increases, the binomial distribution becomes more symmetric and bell-shaped, especially if $p$ is not too close to 0 or 1.

Objective 5: Using the Mean, Standard Deviation, and Empirical Rule to Check for Unusual Results

The empirical rule can be used to determine if an observed result in a binomial experiment is unusual by comparing it to the mean and standard deviation.

• Empirical Rule:

  • Values more than two standard deviations from the mean are considered unusual.

• Example:

  • In a sample of 500 smartphone owners, if 349 use a feature and the mean is $225$with standard deviation $15$, check if 349 is an unusual result.

Additional info: These notes expand on the brief points in the original file, providing full definitions, formulas, and examples for each objective. The table of math symbols is inferred from the context of the original notes.