Probability and Statistics

Learning Objectives

Define probability and calculate the probability of a simple event.

Explain the difference between permutations and combinations.

Calculate the mean, median, and mode of a data set.

The Mathematics of Chance and Data

Probability

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

P(Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

Example: The probability of rolling a 4 on a single standard six-sided die is 1/6, because there is one favorable outcome (rolling a 4) and six total possible outcomes (1, 2, 3, 4, 5, 6).

Permutations vs. Combinations

Both permutations and combinations are ways to count the number of possibilities from a larger set. The key difference is whether order matters.

Permutation: A selection of objects where the order matters. Think of arranging books on a shelf or a race where 1st, 2nd, and 3rd place are distinct.

The number of permutations of k items chosen from a set of n is given by nPk = n! / (n-k)!

Combination: A selection of objects where the order does not matter. Think of choosing a committee of 3 people from a group of 10, or picking lottery numbers.

The number of combinations of k items chosen from a set of n is given by nCk = n! / (k!(n-k)!)

Example: Choosing 3 people from 10.

If they are being awarded Gold, Silver, and Bronze medals, it's a permutation.

If they are being chosen for a committee, it's a combination. There will be fewer combinations than permutations.

Measures of Central Tendency

These are single values that attempt to describe a set of data by identifying the central position within that set.

Mean: The average of the data. It is calculated by summing all the values and dividing by the number of values. It is sensitive to outliers.

Median: The middle value of the data set when it is arranged in order. If there is an even number of values, the median is the average of the two middle values. It is not sensitive to outliers.

Mode: The value that appears most frequently in the data set. A data set can have one mode, more than one mode, or no mode.

Key Terms

Probability: The extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.
Permutation: An arrangement of a set of objects in which the order of the objects is important.
Combination: A selection of items from a collection, such that the order of selection does not matter.
Mean: The average of a set of numbers, calculated by adding all the values and dividing by the count of values.
Median: The value separating the higher half from the lower half of a data sample. It is the middle value when the data is ordered.
Mode: The value that appears most often in a set of data.