Permutations and Combinations

Counting Principles in Probability

Permutations and combinations are fundamental tools in probability and statistics for counting the number of ways events can occur. They are used to calculate probabilities in discrete sample spaces.

• Permutation: The number of ways to arrange n objects taken r at a time (order matters):

• Combination: The number of ways to choose r objects from n (order does not matter):

Example: Choosing 3 students from a class of 10 for a committee (combination), or arranging 3 books on a shelf from 10 (permutation).

The Mean and Standard Deviation of a Discrete Random Variable

Measures of Central Tendency and Spread

For a discrete random variable X with possible values xi and probabilities pi, the mean and standard deviation describe the expected value and variability.

• Mean (Expected Value):

• Standard Deviation:

Example: Calculating the expected number of heads in 3 coin tosses.

Binomial Random Variable

Discrete Probability Distribution

A binomial random variable counts the number of successes in n independent Bernoulli trials, each with probability p of success.

• Probability Mass Function:

• Mean:

• Standard Deviation:

Example: Probability of getting exactly 4 heads in 10 coin tosses with a fair coin.

Normal Distribution

Continuous Probability Distribution

The normal distribution is a continuous probability distribution characterized by its mean μ and standard deviation σ. Many natural phenomena follow a normal distribution.

• Standardization (Z-score): To compare values from different normal distributions, convert to the standard normal using:

Example: Finding the probability that a test score is above 85 when the mean is 80 and standard deviation is 5.

Sampling Distribution of the Sample Mean

Distribution of Sample Means

The sampling distribution of the sample mean describes the distribution of means from all possible samples of size n drawn from a population with mean μ and standard deviation σ.

• Mean of the Sampling Distribution:

• Standard Deviation (Standard Error):

Example: If the population mean is 100 and standard deviation is 15, the standard error for samples of size 25 is .