Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

List the simple events associated with each experiment. Blood tests are given as a part of the admission procedure at the Monterey Garden Community Hospital. The blood type of each patient (A, B, AB, or O) and the presence or absence of the Rh factor in each patient's blood \(\left(\mathrm{Rh}^{+}\right.\) or \(\mathrm{Rh}^{-}\) ) are recorded.

Step-by-Step Solution

Answer

The 8 simple events associated with the blood test experiment are: A\(\mathrm{Rh}^{+}\), A\(\mathrm{Rh}^{-}\), B\(\mathrm{Rh}^{+}\), B\(\mathrm{Rh}^{-}\), AB\(\mathrm{Rh}^{+}\), AB\(\mathrm{Rh}^{-}\), O\(\mathrm{Rh}^{+}\), and O\(\mathrm{Rh}^{-}\).

Step 1: Identify the blood types

There are 4 main blood types: A, B, AB, and O.

Step 2: Identify the Rh factors

There are two Rh factors: \(\mathrm{Rh}^{+}\) (positive) and \(\mathrm{Rh}^{-}\) (negative).

Step 3: List the possible combinations

In order to find the simple events associated with this experiment, we need to combine each blood type with each Rh factor. We can achieve this by creating a list of possible combinations: 1. Blood type A with \(\mathrm{Rh}^{+}\): A\(\mathrm{Rh}^{+}\) 2. Blood type A with \(\mathrm{Rh}^{-}\): A\(\mathrm{Rh}^{-}\) 3. Blood type B with \(\mathrm{Rh}^{+}\): B\(\mathrm{Rh}^{+}\) 4. Blood type B with \(\mathrm{Rh}^{-}\): B\(\mathrm{Rh}^{-}\) 5. Blood type AB with \(\mathrm{Rh}^{+}\): AB\(\mathrm{Rh}^{+}\) 6. Blood type AB with \(\mathrm{Rh}^{-}\): AB\(\mathrm{Rh}^{-}\) 7. Blood type O with \(\mathrm{Rh}^{+}\): O\(\mathrm{Rh}^{+}\) 8. Blood type O with \(\mathrm{Rh}^{-}\): O\(\mathrm{Rh}^{-}\) These are the 8 simple events associated with the blood test experiment at the Monterey Garden Community Hospital.

Key Concepts

Simple Events

In probability theory, a simple event is an outcome that cannot be further broken down into simpler components. It's the most basic possible result in an experiment.
For our example with blood tests, a simple event combines a specific blood type and an Rh factor.
Each unique pairing between these factors—a blood type and an Rh factor—constitutes a single simple event.
This means, with 4 blood types and 2 possible Rh factor options, we have a straightforward mechanism for determining these foundational outcomes.

Blood Type

Blood types are an essential part of blood classifying systems used in hospitals. They help understand how a person's blood behaves and reacts.
The 4 primary blood types are:

• A
• B
• AB
• O

Each person has one of these types based on the presence or absence of certain antigens on the surface of their red blood cells.
Recognizing these types is critical in medical settings, especially for blood transfusions and understanding specific health conditions.

Rh Factor

The Rh factor is another vital classification in blood typing. It determines the presence (+) or absence (-) of a particular protein (known as the RhD antigen) on the surface of blood cells.
Someone who has this protein is Rh positive ( Rh^{+} ), whereas someone who lacks it is Rh negative ( Rh^{-} ).
Understanding the Rh factor is important in pregnancy and transfusion medicine, as mismatched Rh factors can lead to complications.

Combinatorics

Combinatorics is a mathematical principle used to find various ways elements can be arranged or combined.
In this context, it helps us calculate all possible combinations of blood types and Rh factors.
Given:

• 4 blood types (A, B, AB, O)
• 2 Rh factors ( Rh^{+} , Rh^{-} )

We multiply these numbers to find the total combinations: 4 blood types × 2 Rh factors = 8 simple events.
Combinatorics provides a systematic approach to listing outcomes in probability, ensuring none are missed.