Probability Distributions

Introduction to Probability

Probability is the measure of the likelihood that an event will occur. It is not a guarantee but a quantification of uncertainty. For example, a 20% chance of rain does not mean it will definitely rain or not rain; it simply expresses the likelihood.

• Probability values range from 0 (impossible) to 1 (certain).

Random Variables and Probability Distributions

A random variable is a variable whose value is determined by chance. Probability distributions describe the probabilities for each value of a random variable.

• Discrete random variable: Takes countable values (e.g., number of heads in coin tosses).

• Continuous random variable: Takes infinitely many values, not countable (e.g., body temperature).

Requirements for Probability Distributions

• Random variable x is numerical and associated with probabilities.

• The sum of all probabilities is 1:

• Each probability is between 0 and 1:

Example: Probability Distribution for Two Births

Let x = number of females in two births. The probability distribution is:

Probability Histogram

A probability histogram visually represents a probability distribution, with the vertical axis showing probabilities.

Parameters of a Probability Distribution

• Mean (μ):

• Variance (σ²): or

• Standard deviation (σ):

• Expected value (E):

Significant Values and the Range Rule of Thumb

To identify significantly low or high values:

• Significantly low: or lower

• Significantly high: or higher

• Not significant: Between and

Binomial Probability Distributions

Definition and Requirements

A binomial probability distribution arises from a procedure with:

• Fixed number of trials (n)

• Independent trials

• Two possible outcomes per trial (success/failure)

• Constant probability of success (p) and failure (q = 1 - p)

Notation

• n: Number of trials

• x: Number of successes

• p: Probability of success

• q: Probability of failure

• P(x): Probability of exactly x successes

Binomial Probability Formula

The probability of getting exactly x successes in n trials:

Finding Binomial Probabilities Using Technology

Excel and other software can compute binomial probabilities efficiently.

Mean and Standard Deviation of Binomial Distributions

• Mean:

• Variance:

• Standard deviation:

Normal Probability Distributions

Standard Normal Distribution

The standard normal distribution is a bell-shaped, symmetric distribution with mean 0 and standard deviation 1. Probabilities correspond to areas under the curve.

Uniform Distribution

A uniform distribution has values equally spread over the range, resulting in a rectangular graph. The area under the curve equals 1.

Finding Probabilities with z Scores

• z score:

• Use technology or tables to find areas (probabilities) for regions under the normal curve.

Critical Values

Critical values are z scores that separate significant from non-significant results. For a 95% confidence level, the critical value is .

Converting Nonstandard to Standard Normal Distributions

Any normal distribution can be standardized using the z score formula:

Finding Values from Known Areas

Given a probability (area), use the inverse normal function (e.g., Excel's NORM.INV) to find the corresponding x value.

Estimating Parameters and Determining Sample Sizes

Estimating a Population Proportion

• Point estimate: The sample proportion is the best estimate of the population proportion p.

• Confidence interval: A range of values likely to contain the population parameter.

• Margin of error (E):

• Confidence interval for p:

Determining Sample Size for Proportions

• If is known:

• If is unknown:

• Always round up to the next whole number.

Estimating a Population Mean (σ Not Known)

• Point estimate: The sample mean is the best estimate of the population mean μ.

• Confidence interval:

• Margin of error (E): (use t-distribution with df = n - 1)

Determining Sample Size for Means

• If σ is unknown, estimate using range/4 or prior studies.

Summary Table: Choosing the Correct Distribution

Key Terms and Concepts

• Random variable: Variable whose value is determined by chance.

• Probability distribution: Table, formula, or graph showing probabilities for each value of a random variable.

• Binomial distribution: Probability distribution for a fixed number of independent trials with two outcomes.

• Normal distribution: Symmetric, bell-shaped distribution described by mean and standard deviation.

• Confidence interval: Range of values used to estimate a population parameter.

• Margin of error: Maximum likely difference between sample statistic and population parameter.

• Critical value: Value that separates significant from non-significant results in hypothesis testing or estimation.