Question

Two nickels and a dime are shaken together and thrown. We are allowed to keep the coins that turn up heads. We choose a sample space \(\Omega=\\{0,5,10,15,20\\},\) the outcomes of which correspond to the amounts that we can keep. For each of the following situations, cither describe the situation as an event in \(\Omega\) by listing the elements in the appropriate subset of \(\Omega\) or state that the situation cannot be described as an event in this particular sample space: (a) No heads. (b) All heads. (c) Exactly one coin turns up heads. (d) Exactly one of the nickels tums up heads. (e) The dime turns up heads.

Short answer

(a) $\{0\}$, (b) $\{20\}$, (c) $\{5,10\}$, (d) $\{5\}$, (e) $\{10,15\}$.

Step-by-step solution

No Heads Event

For the outcome of no heads, none of the coins show heads. Thus, the amount of money kept is \(0. The event is \)\{0\}$.

All Heads Event

If all coins show heads, we have two nickels and a dime, totaling \(20. The event is \)\{20\}$.

Exactly One Head Event

The outcome has one coin showing heads. This could be either one nickel (\(5) or the dime (\)10), making the event \(\{5,10\}\).

Exactly One Nickel Heads Event

This describes the situation where only one of the two nickels turns up as heads, not the dime. This corresponds to keeping \(5 from one nickel. The event is \)\{5\}$.

Dime Heads Event

If the dime shows heads, then either outcome of \(10 (dime only) or \)15 (dime + 1 nickel) is possible. However, \(15 doesn't solely correspond to just the dime showing heads. So, the event is \)\{10,15\}$.

Key concepts

Discrete Mathematics

Discrete mathematics focuses on the study of distinct and separate values, often encapsulating concepts such as integers, graphs, and combinatorial designs. In this context, we deal with discrete sample spaces, which consist of specific and countable outcomes.
For instance, when tossing a collection of coins, each coin has two possible outcomes - heads or tails. The number of heads showing up in a set of tossed coins represents a value in a discrete sample space. This sample space includes discrete quantities like 0, 5, 10, 15, and 20, each correlating to specific arrangements of heads from the coins.
The scenario described is a prime example of discrete mathematics because it involves analyzing these separate, discrete events rather than dealing with continuous data or ranges. Discrete mathematics provides the tools to understand and work with data that are distinct and unconnected by a continuum.

• Discrete Values: Tied to specific, distinct outcomes.
• Countable Outcomes: Includes a finite set of results, such as those from a set of coin tosses.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and choosing objects systematically. It plays a crucial role in understanding sample spaces and the probabilities of various events occurring. In the coin tossing exercise, combinatorics helps us calculate possible outcomes of coins showing heads.
The problem requires us to determine specific outcomes - these could be situations where no coins show heads, all coins do, or only certain coins do. Each event has a distinct set of possible outcomes. By employing combinatorial techniques, we can determine all these possibilities and represent them in the sample space.

• Counting Structures: Enumerate possible outcomes.
• Arrangement: Consider different formations of heads and tails.
• Selection: Identify scenarios like exactly one coin showing heads.

Through combinatorics, students can better grasp how to systematically approach problems involving selections, arrangements, and distributions, like the one given in the exercise.

Probability Theory

Probability theory is a mathematical framework for quantifying the likelihood of various outcomes. In the context of sample spaces and coin tossing, it helps us understand the probabilities associated with different events occurring.
This exercise teaches us to link outcomes in the sample space with the specific monetary values we might retain. Each event, such as obtaining no heads or exactly one head, forms a subset of this sample space. The ultimate aim is to assign probabilities to these subsets, indicating how likely each scenario is.
Probability theory requires carefully defining the sample space and understanding how to calculate probabilities based on it. For example, when rolling no heads, all tails is a singular outcome in the set, while having exactly one head has multiple coin-flip combinations, impacting its probability.

• Sample Space: Defines potential outcomes.
• Probability Distributions: Assigns likelihood to each event.
• Event Subsets: Each event is a specific subset of the sample space.

By integrating these concepts, learners better comprehend the fundamentals of probability, crucial for analyzing expected outcomes in any stochastic process.