Problem 1 - Calculus for Biology and Medicine

Question

Determine the sample space for each random experiment. The random experiment consisting of tossing a coin three times.

Step-by-Step Solution

Answer

The sample space is S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

Step 1: Understand the Experiment

The experiment involves tossing a single coin three times. A coin has two possible outcomes for each toss: heads (H) and tails (T).

Step 2: Identify the Possible Outcomes for Each Toss

For each individual toss of the coin, there are two possible outcomes: heads (H) or tails (T).

Step 3: Calculate Total Possible Outcomes

Since there are three tosses and each toss has 2 possible outcomes, the total number of possible outcomes is \(2^3 = 8\).

Step 4: List All Possible Outcomes

List down all the combinations possible by considering each toss as a sequence of three results: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Step 5: Verify the Sample Space

Check to ensure all 8 combinations are listed, confirming the complete sample space: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

Key Concepts

Random Experiment

A random experiment is any process that leads to an uncertain result. This idea is fundamental to probability and statistics. It refers to any action or process you perform with unknown outcomes at the start. Each execution of the experiment might produce different results.

For example, when tossing a coin, you perform a random experiment. Each toss is unpredictable even though you may know the possible outcomes ('Heads' or 'Tails'). Despite this uncertainty, you can still measure probabilities by defining the sample space, which includes all potential outcomes.

Random experiments are everywhere. They represent situations where multiple results are possible, and offer a structured way to analyze randomness.

Possible Outcomes

In probability, possible outcomes are the potential results that can occur when a random experiment is performed. It's important to clarify that these "outcomes" are the basic results you might expect from the experiment.

In the example of tossing a coin, each individual toss has two possible outcomes:

• Heads (H)
• Tails (T)

When talking about multiple events such as the toss of a coin three times, the array of outcomes becomes more extensive. For each toss adding in, the possible outcome set doubles.

Hence, when a coin is tossed three times, you have to account for combinations like: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Altogether, there are 8 combinations, representing all possible outcomes of the experiment.

Coin Toss

A coin toss is perhaps the simplest example of a random experiment. It involves flipping a coin and observing the side that lands face up.

Every coin toss essentially has two possible outcomes:

• Heads (H)
• Tails (T)

This straightforward setup makes it a perfect teaching tool for understanding probability, randomness, and sample space. It’s easy to grasp, yet it encapsulates the core principles of taking a random event and deriving meaningful conclusions about potential outcomes.

While tossing a single coin is simple, the complexity increases as you repeat the toss, such as when tossing a coin three times. This repetition allows for a wider range of possibilities—the more you toss, the more complex the outcome list becomes, illustrating how probability works in larger datasets and more complicated scenarios.