Problem 1 - A Graphical Approach to Precalculus with Limits

Question

State a sample space \(S\) with equally likely outcomes for each experiment. A two-headed coin is tossed once.

Step-by-Step Solution

Answer

The sample space is \( S = \{ H \} \).

Step 1: Understanding the Problem

The problem asks us to determine the sample space for an experiment involving a two-headed coin being tossed once. Since the coin is two-headed, it does not have a tail side.

Step 2: Defining the Sample Space

A sample space is a set of all possible outcomes of an experiment. Since the coin has two heads, the only possible outcome of the toss is obtaining a 'Head' since there is no other side.

Step 3: Listing the Sample Space

For the given experiment, the sample space can be listed as: \( S = \{ H \} \), where 'H' represents the outcome of obtaining a head.

Step 4: Verifying Equally Likely Outcomes

Since there is only one outcome in the sample space and the coin always lands on 'Head', all outcomes (just one outcome here) are equally likely.

Key Concepts

Two-Headed Coin

A two-headed coin is a unique or trick coin that has heads on both sides, lacking the typical head and tail. This specific design makes it an intriguing subject when discussing probability and sample spaces. In a standard coin toss with a regular coin, there are two possible outcomes—heads or tails. However, with a two-headed coin, the only possible result of a toss is getting 'heads'.
This peculiar feature is often used in probability problems to illustrate concepts more simply. Again, the absence of a tails side limits the possible number of outcomes to just one, making it a great tool for learning about controlled experiments in probability.

Equally Likely Outcomes

In probability theory, equally likely outcomes refer to the concept that all possible outcomes of an experiment have the same probability of occurring. This is a fundamental principle and is very straightforward when using a two-headed coin.

• Since the coin only has heads, there's no chance of getting anything other than 'heads', making the probability 1 or 100% for that outcome.
• There are no other possibilities, so every "side" of the coin is equally likely, even if it's just one side!

Thus, defining equally likely outcomes in the context of a two-headed coin is simple: there's only one outcome possible, and it is certain.

Experimental Probability

Experimental probability is determined by carrying out an experiment and recording the results, as opposed to theoretical probability, which is based purely on known data and logical deduction. With a two-headed coin, the experimental probability of getting a 'head' is always absolutely predictable.
If you toss the coin several times, you will always record the result 'heads'. The experimental probability, hence, becomes a simple matter of confirming what we already know:

• If you toss it 10 times, you'll get 10 heads, making the experimental probability of heads in this case 1, or 100%.
• Unlike regular coins, further trials do not result in variability since all outcomes are the same.

This example powerfully illustrates how experimental data can confirm theoretical predictions when the conditions leave no room for variance.