Problem 11 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The sample space S is \[ \text{{\bignewline\bigl( H, H \bigr)}, \bigl( H, T \bigr), \bigl( T, H \bigr), \bigl( T, T \bigr) \bignewline\big] \]. Each outcome has a probability of 0.25.

Step 1: Identify the Experiment

The experiment involves tossing a fair coin twice. Each toss has two possible outcomes: Heads (H) or Tails (T).

Step 2: Determine the Sample Space (S)

To list the sample space, consider all possible outcomes of the two coin tosses. Each outcome can be represented as an ordered pair (First Toss, Second Toss).

Step 3: List the Elements of the Sample Space

The sample space S for tossing a coin twice is: \[ S = \text{{\bigewline{\bigl( H, H \bigr)}, \bigl( H, T \bigr), \bigl( T, H \bigr), \bigl( T, T \bigr) \bignewline} \big] \] This set considers all possible combinations of H and T in two tosses.

Step 4: Assign Probabilities to Each Outcome

Since the coin is fair, each of the four outcomes in the sample space is equally likely. The probability of each outcome is therefore calculated as: \[ P(\text{{outcome}}) = \frac{1}{4} = 0.25 \]

Step 5: Construct the Probability Model

The probability model for the experiment is: \[ P \bigl( (H, H) \bigr) = 0.25 \bignewline P \bigl( (H, T) \bigr) = 0.25 \bignewline P \bigl( (T, H) \bigr) = 0.25 \bignewline P \bigl( (T, T) \bigr) = 0.25 \bignewline \]

Key Concepts

Sample Space

When discussing probability, the 'sample space' is a fundamental concept. It represents all possible outcomes of an experiment. For instance, if you toss a fair coin twice, you need to consider every combination of heads and tails.

In this scenario, each coin toss results in either heads (H) or tails (T). When you toss the coin twice, the ordered pairs of outcomes are listed.
To construct the sample space, we denote these pairs as follows: \[ S = \{ (H, H), (H, T), (T, H), (T, T) \} \]

This collection of pairs includes every possible outcome when tossing the coin twice. Understanding the sample space is crucial because it lays the foundation for calculating probabilities.

Fair Coin

A 'fair coin' is a term used frequently in probability. It means that the coin has an equal chance of landing on heads or tails. There are no biases or irregularities. Each side of the coin is equally probable.

Consider the coin toss experiment: The probability of landing on heads (H) is \[ P(H) = 0.5 \] and the same applies to tails (T) \[ P(T) = 0.5 \]

When you toss a fair coin twice, each combination in the sample space discussed earlier holds the same likelihood, thanks to the impartial nature of the coin. This equal probability characteristic is essential for creating accurate probability models.

Equally Likely Outcomes

In a probability model, 'equally likely outcomes' is a core principle. It means each possible result of an experiment has the same chance of occurring.

To understand this concept, let’s revisit the sample space of tossing a fair coin twice. The outcomes \[ (H, H), (H, T), (T, H), (T, T) \] are equally likely.

Because the coin is fair, there's no reason to believe one combination is more likely than another. Therefore, the probability for each is: \[ P(\text{{outcome}}) = \frac{1}{4} = 0.25 \]

Creating the probability model involves assigning these equal probabilities to each outcome:
\[ P((H, H)) = 0.25, \ P((H, T)) = 0.25, \ P((T, H)) = 0.25, \ P((T, T)) = 0.25 \]

Understanding equally likely outcomes helps in creating accurate models to predict and analyze results in probability experiments.