Chapter 4: Probability

Sample Spaces and Basic Probability

The sample space is the set of all possible outcomes of an experiment. An event is one or more outcomes from the sample space. Basic probability is calculated as the ratio of favorable outcomes to total possible outcomes, assuming all outcomes are equally likely.

• Probability Formula:

• Example: A bag contains 5 red and 3 blue balls.

Empirical Probability

Empirical probability is estimated from observed data, rather than theoretical assumptions.

• Formula:

• Example: A player makes 42 of 60 free throws.

Complement Rule

The complement of an event A, denoted as A', is the event that A does not occur. The probability of the complement is one minus the probability of the event.

• Formula:

• Example: If , then

Addition Rule (OR)

The addition rule calculates the probability that at least one of two events occurs. For mutually exclusive events, simply add their probabilities.

• Formula:

• Mutually Exclusive:

• Example: Drawing a king or a heart from a deck:

Multiplication Rule (AND)

The multiplication rule is used to find the probability that two events both occur. If events are independent, multiply their probabilities.

• Formula: ; if independent:

• Example: Rolling a die twice, probability of two 6s:

Conditional Probability

Conditional probability is the probability of event A given that event B has occurred.

• Formula:

• Example: 12 students are athletes; 8 of those play an instrument.

Two-Way Tables

Two-way tables display joint and conditional probabilities for categorical data.

• Joint Probability: Intersection cell / grand total

• Conditional Probability: Intersection cell / total of the given row or column

Mutually Exclusive vs. Independent Events

Mutually exclusive events cannot occur together, while independent events do not affect each other's probabilities.

• Mutually Exclusive:

• Independent:

• Test for Independence: or

• Note: Mutually exclusive events with positive probabilities are not independent.

"At Least One" Probability

The probability of at least one occurrence is the complement of none occurring.

• Formula:

• Example: Rolling a die 3 times,

Chapter 5: Discrete Random Variables and Binomial Probability

Probability Distribution Tables

A probability distribution table lists all possible values of a discrete random variable and their probabilities. The probabilities must sum to 1 and each must be between 0 and 1.

• Valid Distribution: for all x;

• Example Table:

• Example: Flipping two coins, X = number of heads.

Reading Probabilities from a Table

To find probabilities for specific values or ranges, sum the relevant probabilities from the table.

• P(X=a): Probability at a

• P(X≤a): Sum probabilities at or below a

• P(X Sum probabilities strictly below a

• P(X≥a): Sum probabilities at or above a

• P(a≤X≤b): Sum probabilities from a through b, including endpoints

• Example: From the two-coin table,

Mean, Variance, and Standard Deviation of Discrete Random Variables

The mean (expected value), variance, and standard deviation describe the central tendency and spread of a probability distribution.

• Mean:

• Variance:

• Standard Deviation:

• Example: X = 0,1,2 with probabilities 0.25, 0.50, 0.25: , variance = 0.50, SD

Binomial Distribution: Conditions and Formulas

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Conditions (BINS):

  • B: Binary outcomes (success/failure)

  • I: Independent trials

  • N: Fixed number n of trials

  • S: Same probability p of success

• Probability Formula:

• Combination Formula:

• Mean:

• Variance:

• Standard Deviation:

• Example: n = 10, p = 0.70. . Mean = 7; variance = 2.1; SD

Binomial Wording and Calculator Commands

Different phrases correspond to different binomial probabilities. Calculator commands can help compute these values.

• Exactly x: , binompdf(n, p, x)

• At most x: , binomcdf(n, p, x)

• Fewer than x:

• At least x:

• More than x:

• Between a and b inclusive:

• Example: n = 10, p = 0.70.

Chapter 6: Normal Distributions and Sampling

Normal Distribution and Empirical Rule

The normal distribution is a continuous, symmetric, bell-shaped curve. The mean, median, and mode are equal, and the total area under the curve is 1.

• Empirical Rule:

  • About 68% of values within 1 SD of the mean

  • About 95% within 2 SD

  • About 99.7% within 3 SD

• Example: , . Scores 85–115 are within 1 SD, so about 68% of scores fall in this range.

Z-Scores

A z-score measures how many standard deviations a value is from the mean. It standardizes values for comparison.

• Formula:

• Convert back:

• Example: , , .

Normal Probabilities

Normal probabilities are areas under the curve. The cumulative distribution function gives the area to the left of z.

• Less than x:

• Greater than x:

• Between a and b:

• Outside:

• Calculator: normalcdf(lower, upper, , )

• Example: , : , area ;

Finding a Value from a Probability

To find a value corresponding to a given probability (percentile), use the inverse normal function.

• Formula:

• Calculator: invNorm(area to the left, , )

• Example: 90th percentile for , : ,

Sampling Distribution of Sample Mean

The sampling distribution of the sample mean describes the distribution of means from repeated samples. The Central Limit Theorem (CLT) states that for large n, the distribution is approximately normal.

• Mean:

• Standard Error (SE):

• Z-score:

• Example: , , . . For , , probability

Sampling Distribution of Sample Proportion

The sampling distribution of the sample proportion is approximately normal for large n, with mean and standard error as follows.

• Sample Proportion:

• Mean:

• Standard Error:

• Z-score:

• Example: , . . For , , probability

Central Limit Theorem (CLT)

The CLT states that the sampling distribution of the sample mean becomes approximately normal as sample size increases, regardless of population shape, provided n is large enough.

• If population is normal: is normal for any n

• If population is not normal: is approximately normal for large n (often )

• Center: ; Spread:

• Note: CLT applies to sampling distributions, not the population itself

Conditions for Sample Proportions

For the normal approximation to the sampling distribution of sample proportions, certain conditions must be met.

• Check: and

• Require: Random sampling and independence

• Without replacement: of population

Assessing Normality of Population Data

To use normal models, population data should be approximately normal. A histogram should be bell-shaped, symmetric, with one main peak and no severe outliers.

• Strong skewness or outliers: May make normal model inappropriate

Calculator Command Summary

Common calculator commands for probability and statistics:

• Common mistake: For sample means use , not . For sample proportions use .

• For "at least one": Use the complement rule.

Practice Quiz Examples

• Rolling a die once:

• Complement: If ,

• Binomial mean: n = 20, p = 0.30,

• Z-score: , , ,

• Sample mean SE: , ,

• Binomial command for exactly 4 successes: binompdf(n, p, 4)

• Probability of at least one success in 4 trials, p = 0.20: