Question

You work 5 evenings each week at a bookstore. Your supervisor assigns you 5 evenings at random from the 7 possibilities. What is the probability that your schedule does not include working on the weekend?

Short answer

The desired probability is the ratio of the combination for choosing 5 days from 5 weekdays over the combination for choosing 5 days from the entire week. Apply the formula for combinations to do the calculations.

Step-by-step solution

Calculate Total Possible Combinations

First, calculate the total number of ways of assigning 5 evenings from 7 possibilities without any restrictions. This can be done using the formula for combinations which is \( C(n, r) = n! / [(n - r)! * r!] \) where n is the total number of possibilities and r is the number of items to choose. Here, \( n = 7 \) and \( r = 5 \). So, total possible combinations = \( C(7, 5) \).

Calculate Desired Combinations

Next, calculate the number of ways of assigning 5 evenings excluding the weekend. As there are 5 weekdays, the number of ways of choosing 5 days is \( C(5, 5) \).

Calculate the Probability

The probability that the schedule doesn't include the weekend is the ratio of desired combinations to total possible combinations. Therefore, Probability = \( C(5, 5) / C(7, 5) \).

Key concepts

Combinations and Permutations

Understanding combinations and permutations is fundamental in computing various probability scenarios. While both concepts deal with selecting items from a group, they differ in one vital aspect: the order of selection.

A permutation is the arrangement of items in a particular sequence where order matters. Imagine arranging books on a shelf where each sequence of books creates a different aesthetic. On the other hand, a combination involves selecting items without regard to the order. This would be like choosing books to read - once they are chosen, the order in which you read them doesn't matter.

When dealing with combinations, we commonly use the formula \( C(n, r) = \frac{n!}{(n - r)! r!} \), where \( n \) represents the total number of items to choose from, and \( r \) is the number of items to select. Combinations are used in probability when we are interested in the selection of items rather than their arrangement.

Factorial Notation

Factorial notation is a mathematical expression often used in the calculation of combinations and permutations. It is symbolized by an exclamation mark (!) after a number and signifies the product of all positive integers less than or equal to that number.

For example, the factorial of 5, denoted as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\). A special case is \(0!\), which is always equal to 1. This notation is pivotal in simplifying the operations involved in computing probabilities, as it can quickly escalate the number of ways items can be arranged or selected.

Probability Calculation

Probability calculation deals with the likelihood of a particular event happening. It is always a number between 0 and 1, where 0 means the event is impossible, and 1 suggests it is certain to occur.

The basic formula for probability is the number of favorable outcomes divided by the total number of possible outcomes. In our bookstore example, to find the probability of not working on the weekend, we calculate the ratio of the desirable combination (weekdays only) to the total combination (any days). Using the factorial notation and combinations formula, we then compute this probability as \( \frac{C(5, 5)}{C(7, 5)} \). To aid understanding, visualize that each possible schedule is like a ticket in a lottery; the favorable outcomes are the 'winning' schedules that don't include weekends among the total pool of schedules.