Problem 15 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Each outcome has a probability of \(\frac{1}{8}\).

Step 1: Identify Possible Outcomes

Each coin has two possible outcomes: heads (H) or tails (T). For three coins, list all possible combinations of these outcomes.

Step 2: List the Sample Space

Combine the outcomes from Step 1 to form the sample space S. The sample space S includes all possible sequences of heads and tails for three coins.

Step 3: Enumerate Combinations

Write out all combinations: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Step 4: Verify the Sample Space

Count the total number of outcomes to ensure they match the expected number, which is \(2^3 = 8\). Thus, the sample space S is complete.

Step 5: Assign Probabilities

Since each coin is fair, each of the 8 outcomes has an equal probability of occurring. Calculate the probability for each outcome by dividing 1 by the total number of outcomes.

Step 6: Construct the Probability Model

List each outcome with its associated probability. Since there are 8 outcomes and they are all equally likely: \(P(\text{Each Outcome}) = \frac{1}{8}\).

Key Concepts

Tossing Coins Experiment

Tossing coins is a classic probability experiment. It's simple yet powerful for teaching fundamentals. Here's a breakdown:

The experiment involves flipping three fair coins simultaneously. Each coin has two possible outcomes: heads (H) or tails (T).

When we toss three coins, we're observing all combinations of these outcomes. This predictive model helps us understand concepts of randomness and fairness. By counting and verifying the possible outcomes, we build a strong foundation in probability theory.

Sample Space Enumeration

Sample space represents all possible outcomes of an experiment. For our three-coin toss experiment, let's enumerate this.

1. First, list outcomes for each coin.
- Coin 1: H or T
- Coin 2: H or T
- Coin 3: H or T

2. Combine outcomes to create the complete set of possible results. This is our sample space S:
- HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

3. Verify the enumeration by ensuring the count matches expected outcomes using the formula: \(2^3 = 8\). Each combination of H and T for three coins results in eight total outcomes, just as listed. Checking the count is crucial for accuracy.

Equal Probability Assignment

Assigning probabilities is the final core concept. All outcomes in our coin-toss experiment are equally likely because coins are fair.

1. Calculate probability for each outcome: Total possible outcomes = 8
Probability of any specific outcome (e.g., HHH) = \(\frac{1}{8}\)

2. Construct the probability model by listing every outcome with its probability:
- HHH: 1/8
- HHT: 1/8
- HTH: 1/8
- HTT: 1/8
- THH: 1/8
- THT: 1/8
- TTH: 1/8
- TTT: 1/8.

This uniform probability model assures that every outcome has an equal chance. Understanding equal probability helps grasp larger concepts in statistics and real-world applications.