91Ó°ÊÓ

Probability models? In each of the following sítuations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. Remember, a legitimate model need not be a practically reasonable model. If the assignment of probabilities is not legitimate, give specific reasons for your answer. (a) Roll a six-sided die, and record the count of spots on the up-face: $$ \begin{gathered} \mathbf{P}(1)=0 \mathrm{P}(2)=1 / 6 P(3)=1 / 3 P(4)=1 / 3 P(5)=1 / 6 P(6)=0 \\\ P(1)=0 \quad P(2)=1 / 6 \quad P(3)=1 / 3 \\ P(4)=1 / 3 \quad P(5)=1 / 6 \quad P(6)=0 \end{gathered} $$ (b) Deal a card from a shuffled deck: $$ \begin{gathered} \mathrm{P}(\text { clubs })=12 / 52 P(\text { diamonds })=12 / 52 P(\text { hearts })=12 / 52 P(\text { spades })=16 / 52 \\ P(\text { clabs })=12 / 52 \quad P(\text { diamonis })=12 / 52 \\ P(\text { bearts })=12 / 52 \quad P(\text { spades })=16 / 52 \end{gathered} $$ (c) Choose a college student at random and record sex and enrollment status: \(P(\) female full-time \()=0.56 \mathrm{P}(\) male full-time \()=0.44 \mathrm{P}(\) female part- time \()=0.24 \mathrm{P}(\) male part-lime \()=0.17\)

Step by step solution

01

Assess the Sum of Probabilities (a)

For a probability distribution to be legitimate, the sum of all the probabilities must be equal to 1. Here, we add up the probabilities: \( P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 0 + \frac{1}{6} + \frac{1}{3} + \frac{1}{3} + \frac{1}{6} + 0 \). Simplifying, this results in \( 0 + 0.1667 + 0.3333 + 0.3333 + 0.1667 + 0 = 1 \). So, the sum is 1. However, the probabilities for \( P(1) \) and \( P(6) \) are 0, which is unusual for a fair die, but it technically still sums to 1.

02

Evaluate Non-Negativity of Probabilities (a)

Each probability in a legitimate model must be between 0 and 1 inclusive. The given probabilities are: \( 0, \frac{1}{6}, \frac{1}{3}, \frac{1}{3}, \frac{1}{6}, 0 \). All values are within the range from 0 to 1, so this condition is satisfied.

03

Assess the Sum of Probabilities (b)

Again, sum the probabilities: \( P(\text{clubs}) + P(\text{diamonds}) + P(\text{hearts}) + P(\text{spades}) = \frac{12}{52} + \frac{12}{52} + \frac{12}{52} + \frac{16}{52} \). This simplifies to \( \frac{12 + 12 + 12 + 16}{52} = \frac{52}{52} = 1 \). The sum equals 1.

04

Evaluate Non-Negativity of Probabilities (b)

Check if each individual probability is non-negative: \( \frac{12}{52}, \frac{12}{52}, \frac{12}{52}, \frac{16}{52} \) are all positive, satisfying this condition.

05

Assess the Sum of Probabilities (c)

Sum the probabilities for each group: \( P(\text{female full-time}) + P(\text{male full-time}) + P(\text{female part-time}) + P(\text{male part-time}) = 0.56 + 0.44 + 0.24 + 0.17 \). The total is \( 1.41 \), which is greater than 1.

06

Evaluate Non-Negativity of Probabilities (c)

Ensure each probability is non-negative: \( 0.56, 0.44, 0.24, 0.17 \) are all positive, so this condition is met. However, the total exceeding 1 invalidates the model.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution

Probability distribution is a fundamental concept in the study of probability. It describes how probabilities are assigned to each possible outcome in a sample space. Imagine rolling a six-sided die. The probability distribution would assign each side of the die a probability value, representing the likelihood of each side landing face up.

In any legitimate probability distribution, two key conditions must always be met:

• The sum of probabilities of all possible outcomes must be equal to 1. For example, if you're rolling a standard die, the sum of probabilities for landing on one particular side: 1/6 for each of the six sides, should sum up to 1 (i.e., 6/6).
• Each probability value has to lie between 0 (never happens) and 1 (definitely happens), including 0 and 1 themselves. This guarantees that probabilities remain realistic and bounded.

Understanding these requirements helps ensure that the assignment of probabilities is sensible and conforms to the rules of probability. This is vital for creating and analyzing any probability distribution correctly.

Probability Model

A probability model is essentially a mathematical representation of a random phenomenon. It consists of two parts: the sample space, which includes all possible outcomes, and a method for assigning probabilities to events within this sample space. Consider a deck of cards as an example. A probability model would document each card's likelihood of being drawn—such as the probability of drawing a club, a diamond, a heart, or a spade.

The construction of a probability model needs to address the same foundational rules applicable to probability distributions: the total probability of all possible outcomes must sum to 1, and every individual probability must be non-negative, ranging between 0 and 1.

Crafting a probability model allows statisticians and scientists to predict and analyze potential outcomes. It is key, for example, in assessing the fairness of a game or any random process-driven scenario. While a probability model needs to obey mathematical laws, a legitimately constructed model might not always reflect reality or practical situations precisely.

Legitimacy of Probability

For a probability assignment to be considered legitimate, it must strictly adhere to the two essential rules of probability. Firstly, the total probability of all events or outcomes must add up to 1. This ensures that some outcome in the sample space will occur when an event happens. Take, for instance, dealing a card from a standard deck; the probabilities of each suit adding up to 52/52 satisfies this condition.

Secondly, each probability assigned cannot be negative, and it must not exceed 1. Each probability must realistically mirror the chance of its corresponding outcome occurring. For instance, rolling a fair die with probabilities of 0 for some sides still might be mathematically allowed if they sum to 1, but it doesn't accurately reflect fair play.

The exercise illustrates that without meeting these conditions, mathematical representations of probability can't be legitimized. When probabilities do not sum to 1, such as in situation (c) where the total exceeds 1, the model is invalidated. This could highlight either an error in calculations or an inaccurate representation of potential outcomes. These principles guide the creation, assessment, and improvement of probability models.