Problem 16 - Algebra and Trigonometry

Question

List the sample space S of each experiment and (b) construct a probability model for the experiment. Tossing one fair coin three times

Step-by-Step Solution

Answer

Sample space: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Each outcome has a probability of 1/8.

Step 1: Understand the Experiment

The experiment involves tossing one fair coin three times. Each toss can result in either Heads (H) or Tails (T).

Step 2: Determine the Sample Space (S)

List all possible outcomes of the experiment by considering all combinations of H and T for three tosses. For example, HHH, HHT, HTT, etc. The sample space S includes all 8 possible outcomes: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Step 3: Verify the Sample Space

Ensure the sample space includes all possible outcomes: There are 2 outcomes for each toss and 3 tosses, so there should be a total of 2^3 = 8 outcomes, which matches the number of outcomes listed in the sample space.

Step 4: Construct the Probability Model

Since the coin is fair, the probability of getting either Heads or Tails on each toss is equal. Therefore, each outcome in the sample space has an equal probability of occurring. There are 8 possible outcomes, so the probability for each outcome is: P(outcome) = 1/8. The probability model is: P(HHH) = 1/8, P(HHT) = 1/8, P(HTH) = 1/8, P(HTT) = 1/8, P(THH) = 1/8, P(THT) = 1/8, P(TTH) = 1/8, P(TTT) = 1/8.

Key Concepts

Sample Space

In probability, the term 'sample space' refers to the set of all possible outcomes of an experiment. When you toss a fair coin three times, you're conducting an experiment where each toss can either be Heads (H) or Tails (T). To determine the sample space, we need to list every possible combination of these outcomes.

Imagine each coin toss as a decision point with two options: H or T. Repeating this process for three tosses gives us the following combinations: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. The sample space (S) is simply the set of all these combinations:

• HHH
• HHT
• HTH
• HTT
• THH
• THT
• TTH
• TTT

It's essential for the sample space to include all 8 possible outcomes because each of the three coin tosses has 2 outcomes, leading to a total of \(2^3 = 8\) combinations.

Fair Coin

A 'fair coin' means that the coin has no bias; each side of the coin has an equal chance of landing face up. In a fair coin, the probabilities are evenly distributed. This is important when analyzing probabilities because it ensures that each possible outcome from our sample space is equally likely.

For example, when you toss a fair coin three times, the chance of getting Heads or Tails on any single toss remains consistent at 50% (or 0.5). That means that each sequence of results (like HHH or THT) should also be equally probable when calculating the probability model for the entire experiment.

Equal Probability

When we talk about 'equal probability,' we mean that every outcome in the sample space has the same chance of occurring. For our coin-tossing experiment, because the coin is fair, each of the 8 results we've listed has an equal chance of happening.

To quantify this, we calculate the probability of each outcome by dividing 1 (the total probability) by the number of possible outcomes in the sample space. Since there are 8 outcomes, each outcome has a probability of:

\[ \text{P(outcome)} = \frac{1}{8} = 0.125 \]

This equal probability ensures that no single outcome is more likely than another, maintaining the principle of fairness in the experiment. Hence, every outcome like HHH, HHT, and so on, each would have a probability of 0.125.

Probability Distribution

A 'probability distribution' lists all potential outcomes of an experiment along with their corresponding probabilities. For our example, tossing a fair coin three times, the probability distribution assigns a probability of 0.125 to each of the 8 outcomes in the sample space.

The probability distribution for our coin tosses is as follows:

• P(HHH) = 1/8
• P(HHT) = 1/8
• P(HTH) = 1/8
• P(HTT) = 1/8
• P(THH) = 1/8
• P(THT) = 1/8
• P(TTH) = 1/8
• P(TTT) = 1/8

Each outcome is equally likely, making this a uniform probability distribution. Understanding this distribution is key to solving probability problems, as it helps visualize and quantify the likelihood of each possible outcome in the experiment.