Question

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

Short answer

Sample Space S = {HH, HT, TH, TT}; P(HH) = P(HT) = P(TH) = P(TT) = 1/4.

Step-by-step solution

- Define the Experiment

The experiment involves tossing two fair coins once.

- Identify Possible Outcomes

Each coin can land on heads (H) or tails (T). Since two coins are tossed, the possible outcomes are: 1. Both coins show heads (HH)2. The first coin shows heads and the second shows tails (HT)3. The first coin shows tails and the second shows heads (TH)4. Both coins show tails (TT)

- List the Sample Space

The sample space, S, is the set of all possible outcomes: S = {HH, HT, TH, TT}

- Determine Probabilities of Each Outcome

Since the coins are fair, each outcome is equally likely. There are 4 possible outcomes, so the probability of each outcome is P(Outcome) = 1/4.

- Construct the Probability Model

List each outcome with its corresponding probability: P(HH) = 1/4P(HT) = 1/4P(TH) = 1/4P(TT) = 1/4

Key concepts

Sample Space

In probability theory, the 'sample space' refers to the set of all possible outcomes of an experiment. For instance, when tossing two coins, the sample space includes all the combinations of heads (H) and tails (T) that the coins can land on. Each coin can land in one of two states—heads or tails—leading to a specific combination for each toss.

Hence, when tossing two fair coins, the sample space S is:

• HH (both coins show heads)
• HT (the first coin shows heads, and the second shows tails)
• TH (the first coin shows tails, and the second shows heads)
• TT (both coins show tails)

Understanding the 'sample space' is crucial because it helps in determining the likelihood of different outcomes. Each outcome is unique in terms of how the coins land.

Fair Coin

A 'fair coin' is a coin that has an equal probability of landing on either heads or tails. This means that there is no bias; each side has a 50% chance of appearing when tossed. Fair coins are fundamental in probability experiments because they ensure that the only factor affecting the outcome is chance.

For example, when tossing a fair coin, the probabilities are:

• Heads (H) = 0.5
• Tails (T) = 0.5

When dealing with fair coins, calculations about likelihoods become straightforward. This makes them suitable for simple probability models and educational exercises.

Probability Outcome

The term 'probability outcome' pertains to the likelihood of a specific result occurring within a given sample space. For example, in the context of tossing two fair coins, there are four possible outcomes (HH, HT, TH, TT), and each is equally likely.

Because the coins are fair, the probability of each specific outcome can be calculated by recognizing that each outcome has an equal chance of occurring. Given four possible outcomes, the probability of each outcome is calculated as:
\[P(Outcome) = \frac{1}{4}\]
Thus,

• P(HH) = \( \frac{1}{4} \)
• P(HT) = \( \frac{1}{4} \)
• P(TH) = \( \frac{1}{4} \)
• P(TT) = \( \frac{1}{4} \)

This demonstrates that each outcome in the sample space has a 1 in 4 chance of occurring.

Tossing Coins

Tossing coins is one of the simplest yet most illustrative experiments in probability theory. It involves flipping one or more coins and observing the outcome. Every flip is an independent event where previous flips do not affect future ones.

When tossing two coins, you consider each coin's result independently. The combined results constitute the outcome of a single toss of the two coins. The combined results from this experiment are compiled to form the sample space for analysis.

Here’s why this simple experiment is so effective for learning probability:

• The outcomes are easy to understand and visualize.
• It demonstrates core concepts like independent events and equal likelihood.
• It allows for easy calculation of probabilities, making it an excellent teaching tool.

Tossing coins demonstrates how probability models work and helps in understanding larger, more complex probability problems.