Question

In Exercises \(\mathrm{P} .63\) to \(\mathrm{P} .67,\) state whether the process described is a discrete random variable, is a continuous random variable, or is not a random variable. Draw one M\&M from a bag. Observe whether it is blue, green, brown, orange, red, or yellow.

Short answer

The process of drawing one M&M from a bag and observing the color is a discrete random variable.

Step-by-step solution

Understand the Problem

Recognize that the problem statement describes a scenario: drawing one M&M from a bag and observing the color. The colors that can be observed are blue, green, brown, orange, red, or yellow. These are distinct and countable outcomes.

Define the Types of Random Variables

Identify the types of variables to distinguish between. There are two types of random variables: discrete and continuous. A discrete random variable has a countable number of possible values. A continuous random variable, however, has an infinite number of possible outcomes.

Classify the Random Variable

Since the colors of the M&M's are countable and distinct, it can be inferred that each draw from the bag will result in one of these outcomes, and no other. Therefore, the process can be classified as a discrete random variable.

Key concepts

Types of Random Variables

Understanding the types of random variables is crucial when dealing with probability and statistics. Let's dive into the two main categories of random variables: discrete and continuous.

A discrete random variable is characterized by a countable number of distinct outcomes. Think of it like a playlist with a limited set of songs – you can count and list each one. Examples include rolling a six-sided die (which can result in 1, 2, 3, 4, 5, or 6), or the number of heads when flipping a coin multiple times.

On the other hand, a continuous random variable has an infinite number of possible outcomes that can't be counted but can be measured. Imagine measuring the time it takes to run a marathon – the result can be any time from several hours to the exact millisecond, forming an uncountable set.

When you encounter a problem, like the one involving M&Ms, you're often asked to classify the random variable involved. Is it discrete, with a set number of outcomes, or is it continuous, with a range of possibilities? Recognizing the difference between these helps you determine how to approach probability calculations, as the methods differ for each type.

Continuous Random Variable

Focusing more deeply on the continuous random variable, let's consider its characteristics and implications in the study of probability.

Variables of this kind represent outcomes that are not just numerous but form a continuum. Think of measuring temperature on a thermometer. It could be 73 degrees, 73.1 degrees, 73.11 degrees, and so on – an infinite sequence of potential temperatures could occur.

Probability calculations for continuous random variables often involve concepts like probability density functions (pdf) and cumulative distribution functions (cdf). These allow us to find the likelihood of a variable falling within a certain range of values, rather than pinpointing an exact outcome, which is impossible due to the infinite nature of such variables.

For example, if we measure how much rain falls during a day, we might say the probability of getting more than an inch is 30%, rather than trying to calculate the probability for exactly one inch of rain, which would be practically zero in a continuous setting.

Probability

Probability is the cornerstone of random variables and essentially quantifies the chances of an event occurring.

For any random variable, whether discrete or continuous, probability tells us how likely it is for a specific outcome or range of outcomes to happen. It's expressed as a number between 0 and 1, with 0 meaning the event will not occur, and 1 signifying certainty that the event will occur.

The basic formula for probability is:
\[\begin{equation}P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\end{equation}\]
For a discrete random variable, like the M&M example, we might ask: 'What's the probability of drawing a blue M&M?' If there are an equal number of each color, and there are six colors, the probability would be \frac{1}{6}\.

However, for a continuous random variable, we can't talk about the probability of a single exact value. Instead, we discuss the probability of the variable falling within a range. For instance, 'What is the probability that a randomly selected day in July will have temperatures between 75 and 85 degrees Fahrenheit?'. To answer this, we would use probability density functions, integrating over the range of interest.

In both cases, the concept of probability allows us to make informed predictions and assessments of various scenarios, playing a vital role in decision-making and analysis in various fields.