Lesson 10.1: Probability and Sample Spaces

Introduction

In this lesson, we will explore the fundamental concepts of probability, an essential aspect of decision-making in business, science, and many areas of social science. Our objectives are to understand outcomes, events, and sample spaces, as well as to learn about the probability scale and equally likely outcomes. We will also discover how to estimate probability from relative frequency, list the sample space for various experiments, and calculate probabilities based on equally likely outcomes.

Objectives:

Define outcomes, events, and sample spaces.
Understand the probability scale and the concept of equally likely outcomes.
Estimate probability from relative frequency data.
List the sample space for a simple experiment.
Calculate probability from equally likely outcomes.

What is Probability?

Probability is a branch of mathematics that deals with the likelihood of events occurring. It is a way of quantifying uncertainty and helps in making informed decisions based on the chances of various outcomes. The probability of an event is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. The probability $ P(E) $ of an event $ E $ can be expressed mathematically as follows:

$$ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} $$

Outcomes and Events

An outcome is the result of a single trial of a random experiment. For instance, when flipping a coin, there are two possible outcomes: heads (H) or tails (T). An event, on the other hand, is a collection of one or more outcomes. For example:

The event of getting heads when flipping a coin can be noted as $ E_1 = \{H\} $.
The event of getting tails can be noted as $ E_2 = \{T\} $.
The combined event of obtaining either heads or tails is noted as $ E = \{H, T\} $.

Sample Space

The sample space, denoted as $ S $, is the set of all possible outcomes of a random experiment. For our coin-flipping example, the sample space is defined as:

$$ S = \{H, T\} $$

To illustrate further, let’s consider rolling a six-sided die. The possible outcomes when rolling the die are the numbers 1 through 6. Therefore, the sample space in this case is:

$$ S = \{1, 2, 3, 4, 5, 6\} $$

Worked Example: Sample Space

Let's take a more complex example by examining the experiment of drawing a card from a standard deck of 52 playing cards.

Sample Space:

The sample space $ S $ consists of all the cards in the deck, which includes:

$$ S = \{ \text{Ace of Hearts, 2 of Hearts, 3 of Hearts, ..., King of Spades} \} $$

Events:

If we define event $ E_1 $ as drawing a heart, the event can be represented as follows:

$$ E_1 = \{ \text{Ace of Hearts, 2 of Hearts, 3 of Hearts, ..., King of Hearts} \} $$

This event consists of 13 favorable outcomes since there are 13 hearts in the deck.

The Probability Scale

The probability of any event can be quantified on a scale from 0 to 1, where:

A probability of 0 means the event is impossible; it cannot happen.
A probability of 1 means the event is certain; it will definitely occur.
Intermediate values indicate varying degrees of likelihood.

Equally Likely Outcomes

When all outcomes in the sample space are equally likely, calculating probability becomes straightforward. If an outcome $ E $ can occur in $ n $ different ways, the probability $ P(E) $ can be computed as follows:

$$ P(E) = \frac{\text{Number of favorable outcomes}}{n} $$

Worked Example: Equally Likely Outcomes

Consider the scenario of rolling a fair six-sided die. Each side of the die is equally likely to land face up, so the total number of outcomes $ n $ is:

$$ n = 6 $$

If we want to find the probability of rolling a 3, the number of favorable outcomes is just 1 (rolling a 3), thus:

$$ P(\text{rolling a 3}) = \frac{1}{6} $$

Conversely, if we wanted to calculate the probability of rolling an even number (the outcomes are 2, 4, and 6, making 3 favorable outcomes), we would find:

$$ P(\text{rolling an even number}) = \frac{3}{6} = \frac{1}{2} $$

Estimating Probability from Relative Frequency

Often, especially in practical applications, we may not know the total number of possible outcomes. In such cases, we can estimate the probability of an event by using relative frequency. The relative frequency is calculated using the formula:

$$ \text{Relative Frequency} = \frac{\text{Number of times event occurs}}{\text{Total number of trials}} $$

Then, the estimated probability can be denoted as:

$$ P(E) \approx \text{Relative Frequency} $$

Worked Example: Estimating Probability

Let’s say we flip a coin 100 times and count the number of heads that appear. If heads show up 55 times, the relative frequency of obtaining heads is:

$$ \text{Relative Frequency of Heads} = \frac{55}{100} = 0.55 $$

We can estimate the probability of flipping heads as:

$$ P(\text{Heads}) \approx 0.55 $$

Similarly, if tails show up 45 times, we calculate:

$$ P(\text{Tails}) \approx \frac{45}{100} = 0.45 $$

Conclusion

In this lesson, we explored probability, sample spaces, outcomes, events, and the probability scale. We learned how to calculate the probability of events using both theoretical and empirical methods. Understanding these foundational concepts prepares us for more advanced topics in probability, such as conditional probability and discrete probability distributions in future lessons.

Study Notes

Probability quantifies the likelihood of events occurring.
An outcome is a single result of an experiment; an event is a collection of outcomes.
The sample space is the set of all possible outcomes of an experiment.
Probabilities range from 0 (impossible event) to 1 (certain event).
Equally likely outcomes simplify probability calculations.
Relative frequency can be used to estimate probabilities when outcomes are not fully known.