Chapter 6 - Discrete Probability Distributions and Binomial Experiments

Discrete Probability Distributions

Random Variables

A random variable (RV) is a numerical measure of the outcome of a probability experiment. The value of a random variable is determined by chance.

• Discrete Random Variable: Takes on a finite or countable number of values. Examples include the number of heads in a series of coin tosses.

• Continuous Random Variable: Takes on an infinite or uncountable number of values, often measured rather than counted (e.g., height, weight).

Example: Let X be the random variable representing the number of tails in a coin toss experiment with n = 100 trials.

Probability Distribution

A probability distribution for a discrete random variable X lists each possible value of X together with its probability.

• Each probability P(X) must satisfy $0 \leq P(X) \leq 1$.

• The sum of all probabilities must equal 1: $\sum P(X) = 1$.

Example: Probability distribution for the number of heads in 100 coin tosses.

Mean (Expected Value) of a Probability Distribution

The mean or expected value of a discrete probability distribution is calculated as:

• $\mu = E(X) = \sum x_i P(x_i)$

This represents the long-run average outcome of the random variable.

Standard Deviation of a Probability Distribution

The standard deviation

• $\sigma = \sqrt{\sum (x_i - \mu)^2 P(x_i)}$

This quantifies the variability of the distribution.

Binomial Probability Experiments

Definition and Properties

A binomial experiment is a specific type of discrete probability experiment that satisfies four conditions:

• Fixed number of trials (n): The experiment is repeated a set number of times (e.g., n = 100 coin tosses).

• Independent trials: The outcome of one trial does not affect the outcome of another.

• Two mutually exclusive outcomes: Each trial results in either a success or a failure.

• Constant probability of success (p): The probability of success remains the same for each trial.

Example: Tossing a fair coin 100 times, where 'heads' is considered a success and 'tails' a failure. Probability of success (heads) is $p = 0.5$ for each toss.

Mean and Standard Deviation of a Binomial Random Variable

The mean and standard deviation for a binomial random variable are given by:

• Mean (Expected Value): $\mu = np$

• Standard Deviation: $\sigma = \sqrt{npq}$, where $q = 1 - p$ is the probability of failure.

Example: For n = 100 coin tosses, p = 0.5: Mean: $\mu = 100 \times 0.5 = 50$ Standard Deviation: $\sigma = \sqrt{100 \times 0.5 \times 0.5} = \sqrt{25} = 5$

Empirical Rule (Additional info)

The Empirical Rule applies to normal distributions and states that:

• Approximately 68% of data falls within 1 standard deviation of the mean.

• Approximately 95% within 2 standard deviations.

• Approximately 99.7% within 3 standard deviations.

This rule is useful for understanding the spread of data in binomial experiments when n is large and p is not too close to 0 or 1.

Summary Table: Binomial Experiment Properties

Property	Description	Formula
Number of Trials	Fixed, denoted by n	n
Probability of Success	Constant for each trial	p
Probability of Failure	Complement of success	q = 1 - p
Mean	Expected number of successes	$\mu = np$
Standard Deviation	Spread of number of successes	$\sigma = \sqrt{npq}$

Applications and Examples

• Analyzing the results of coin toss experiments using pie charts to visualize proportions of outcomes.

• Using software tools (e.g., StatCrunch) to organize and summarize data from binomial experiments.

• Calculating expected values and standard deviations to interpret the results of repeated trials.

Additional info: Pie charts are often used to visually represent the proportion of successes and failures in binomial experiments. For large n, the binomial distribution approximates the normal distribution, allowing the use of the Empirical Rule.