Probability Distributions and Binomial Probability: Study Notes for Statistics Exam Prep

Probability Distributions

Probability Distribution for a Random Variable

A probability distribution describes how probabilities are assigned to each possible value of a random variable. It can be constructed from an experiment (such as tossing a coin three times) or from a frequency distribution.

• Probability Distribution Table: List all possible outcomes and their associated probabilities. Probabilities must be between 0 and 1, and the sum of all probabilities must equal 1.

• Validity of a Probability Distribution: Check that all probabilities are valid (between 0 and 1) and that their sum is exactly 1.

• Example: Tossing a coin three times. Possible outcomes for number of heads: 0, 1, 2, 3. Calculate probability for each outcome.

Formula:

• For discrete random variable $X$ with possible values $x_1, x_2, ..., x_n$ and probabilities $P(x_1), P(x_2), ..., P(x_n)$:

$\sum_{i=1}^{n} P(x_i) = 1$

Mean and Standard Deviation of a Probability Distribution

The mean (expected value) and standard deviation measure the center and spread of a probability distribution.

• Mean (Expected Value): $\mu = \sum x \cdot P(x)$

• Standard Deviation: $\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(x)}$

• Application: Used to identify significantly high or low values in a distribution.

Significantly High and Low Values

Values are considered significantly high or significantly low if they are far from the mean, typically more than two standard deviations above or below the mean.

• Rule of Thumb: Values more than two standard deviations from the mean are considered unusual.

• Cutoff Values: $\text{Significantly high: } x > \mu + 2\sigma$ $\text{Significantly low: } x < \mu - 2\sigma$

• Probability Criteria: Outcomes with probability less than 0.05 are considered unusual.

Expected Value in Games

The expected value is used to determine the average outcome of a game or experiment over many repetitions.

• Formula: $E(X) = \sum x \cdot P(x)$

• Application: Used to analyze the cost or payoff in games of chance.

• Example: Calculating the expected winnings in a lottery game.

Binomial Probability Distributions

Recognizing Binomial Experiments

A binomial experiment is a statistical experiment with the following properties:

• Fixed number of trials ($n$)

• Each trial has two possible outcomes: success or failure

• Probability of success ($p$) is the same for each trial

• Trials are independent

Example: Flipping a coin 10 times and counting the number of heads.

Binomial Probability Formula

The probability of getting exactly $k$ successes in $n$ independent trials is given by:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

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

• $p$ is the probability of success

• $1-p$ is the probability of failure

Using Technology for Binomial Probabilities

Calculators (such as the TI-83/84) can be used to compute binomial probabilities efficiently. Use the binomial probability function to find the probability of a specific number of successes.

• TI-83/84: Use binompdf for exact probabilities, binomcdf for cumulative probabilities.

• Manual Calculation: Use the binomial formula above.

Mean and Standard Deviation of Binomial Distribution

The mean and standard deviation for a binomial distribution are calculated as:

• Mean: $\mu = np$

• Standard Deviation: $\sigma = \sqrt{np(1-p)}$

Significantly High and Low Values in Binomial Distributions

Similar to general probability distributions, use the mean and standard deviation to identify significantly high or low numbers of successes.

• Significantly high: Number of successes $> \mu + 2\sigma$

• Significantly low: Number of successes $< \mu - 2\sigma$

• Probability criteria: Probability less than 0.05 is considered unusual.

Normal Probability Distributions

Finding Probabilities for Normal Random Variables

For a normal distribution, probabilities are found for values above, below, or between given values using the mean and standard deviation.

• Standardization: Convert values to z-scores using $z = \frac{x - \mu}{\sigma}$

• Use of Tables: Use standard normal tables or graphing calculators to find probabilities.

• Example: Probability that a value is greater than 100 when $\mu = 90$ and $\sigma = 10$.

Cutoff Values and Percentiles

Find cutoff values (such as the value that separates the top 10% of a distribution) using z-scores and normal tables.

• Percentile Calculation: Find the z-score corresponding to the desired percentile, then solve for $x$.

• Formula: $x = \mu + z\sigma$

Sample Mean Probabilities

When working with sample means, use the standard deviation of the sample mean:

• Standard Error: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Apply normal distribution methods to sample means.

Summary Table: Key Formulas

Distribution	Mean ($\mu$)	Standard Deviation ($\sigma$)	Probability Formula
Discrete Probability	$\mu = \sum x \cdot P(x)$	$\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(x)}$	Sum of probabilities for each outcome
Binomial	$\mu = np$	$\sigma = \sqrt{np(1-p)}$	$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
Normal	$\mu$	$\sigma$	Use z-score: $z = \frac{x - \mu}{\sigma}$
Sample Mean	$\mu_{\bar{x}} = \mu$	$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$	Use z-score for sample mean

Additional info:

• Some context and definitions were inferred from standard statistics curriculum, as the original notes were fragmented.

• Examples and formulas were expanded for clarity and completeness.