Question

The waiting room of a dentist's office contains a stack of 10 old magazines. During the course of a morning, four patients, who are waiting during non- overlapping times, select a magazine at random to read. Calculate in two ways the probability that two or more patients select the same magazine.

Short answer

The probability that two or more patients select the same magazine is 0.496.

Step-by-step solution

Determine Total Possible Selections

First, identify the number of ways in which the four patients can select a magazine. Each patient has 10 magazines to choose from. Therefore, the total number of possible magazine selections is given by multiplying the choices for each patient: $10\times 10\times 10\times 10={10}^4=10,000$.

Derive Probability of All Unique Selections

Consider the case where each patient selects a unique magazine. The number of ways this can happen is calculated as follows: The first patient has 10 choices, the second patient has 9 remaining unique selections, the third has 8, and the fourth has 7. Thus, the number of ways to select magazines such that no magazine is repeated is $10\times 9\times 8\times 7$.

Calculate Probability of All Different Selections

The probability that all four patients select different magazines is the number of ways to make unique selections divided by the total possible selections: $\frac{10\times 9\times 8\times 7}{{10}^4}=\frac{5040}{10000}=0.504$.

Calculate Probability Using Complementary Approach

Since we need the probability that two or more patients select the same magazine, we use the complementary probability. Subtract the probability that all selections are different from 1: $$1-0.504=0.496$$.

Verify with Complementary Analysis

To double-check, consider all outcomes where at least one magazine is chosen by multiple patients. This is, by definition, 1 minus the probability calculated for all unique selections. Thus, confirm: $1-\frac{5040}{10000}=0.496$.

Key concepts

Combinatorics

Combinatorics is a fascinating branch of discrete mathematics that deals with counting, arranging, and studying patterns of elements. In our exercise, we use combinatorial techniques to determine how many different ways four patients can select magazines. This process involves calculating permutations, which are arrangements of objects in a specific order.
In the case of our problem, each patient selecting from 10 magazines is a sequential choice. The total number of ways the magazines can be selected by the four patients is found by multiplying the number of choices for each patient, i.e., 10 choices per patient, four times. This results in a total of $10\times 10\times 10\times 10={10}^4=10,000$ total combinations.
Combinatorics helps simplify complex scenarios into manageable calculations. It enables us to approach problems methodically, ensuring we account for all possible outcomes. Whether in games, scheduling, or in the waiting room scenario, combinatorics provides valuable insights into the structure of choices available, ensuring no possibilities are overlooked.

Unique Selections

Unique selections occur when each choosing entity—like our patients—selects distinct items without overlap. The challenge in our exercise is determining how many ways the patients can select different magazines. This concept requires understanding permutations further as it involves sequentially reducing the available choices.
For the first patient, all 10 magazines are potential choices. Yet, as each subsequent patient makes a selection, the number of available unique magazines decreases: 9 choices for the second patient, 8 for the third, and 7 for the fourth. Hence, the computation for the unique selections is carried out through the product $$10\times 9\times 8\times 7$$.
This approach ensures no magazine is selected twice, adhering to the principle of one-of-a-kind choice per patient. The total number reflects how without replacement, options quickly diminish. Understanding unique selections is crucial for scenarios requiring non-repetitive choices across multiple actors.

Complementary Probability

Complementary probability is a clever tactic often used to solve probability questions by focusing on what you aren't interested in to find what you are interested in. In our exercise, we seek the probability that at least two patients select the same magazine. Instead of listing each overlapping scenario, it’s more efficient to consider the opposite: the likelihood all patients pick different magazines.
Thus, once we calculate the probability that all selections are unique— $\frac{5040}{10000}=0.504$—we can simply subtract this from 1. This gives us the complementary probability of at least one magazine being selected by multiple patients, found as: $$1-0.504=0.496$$.
Complementary probability elegantly bypasses the complexity of directly addressing all likely overlaps by refocusing on the simpler, all unique scenario. This technique is widely applicable and often simplifies the process of finding the desired probability in various problems.