91Ó°ÊÓ

Probability models? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. If not, give specific reasons for your answer. (a) Roll a 6-sided die and record the count of spots on the up-face: \(P(1)=0, P(2)=1 / 6, P(3)=1 / 3, P(4)=$$$1 / 3, P(5)=1 / 6, P(6)=0$$(b) Choose a college student at random and record gender and enrollment status: \)P(\( female full-time \))=0.56\(, \)P(\( male full-time \))=0.44, P(\( female part-time \))=0.24\( \)P(\( male part-time \))=0.17\( (c) Deal a card from a shuffled deck: \)P(\( clubs \))=12 / 52\(, \)P(\( diamond \)s)=12 / 52, P(\( heart \))=12 / 52, P(\( spades \))=\( \)16 / 52$

Step by step solution

01

Evaluate the Die Roll Probabilities

For a 6-sided die, the probability of each face appearing must be between 0 and 1, and the total probabilities for all outcomes must sum to 1. The given probabilities are \( P(1) = 0, P(2) = \frac{1}{6}, P(3) = \frac{1}{3}, P(4) = \frac{1}{3}, P(5) = \frac{1}{6}, P(6) = 0 \). Sum these probabilities: \( 0 + \frac{1}{6} + \frac{1}{3} + \frac{1}{3} + \frac{1}{6} + 0 = 1 \). This sum is equal to 1, and all probabilities are within the valid range, so it is legitimate.

02

Evaluate Student Enrollment Probabilities

The probabilities assigned to different student enrollments are \( P(\text{female full-time})=0.56 \), \( P(\text{male full-time})=0.44 \), \( P(\text{female part-time})=0.24 \), and \( P(\text{male part-time})=0.17 \). The sum is \( 0.56 + 0.44 + 0.24 + 0.17 = 1.41 \), which is greater than 1. Therefore, this assignment is not legitimate since probabilities cannot sum to more than 1.

03

Evaluate the Card Deck Probabilities

Consider the probabilities for drawing a card from a deck: \( P(\text{clubs}) = \frac{12}{52}, P(\text{diamonds}) = \frac{12}{52}, P(\text{hearts}) = \frac{12}{52}, P(\text{spades}) = \frac{16}{52} \). Summing these: \( \frac{12}{52} + \frac{12}{52} + \frac{12}{52} + \frac{16}{52} = \frac{52}{52} = 1 \). Although the sum equals 1, each type of card should have an equal probability of \( \frac{13}{52} \) since there are 13 cards per suit, so this distribution is not legitimate.

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Key Concepts

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

Rules of Probability

When we discuss probability, we are referring to the likelihood of an event occurring. The rules of probability help us assign these likelihoods in a sensible way. Two primary conditions must be met:

• Each probability must fall between 0 and 1. This means an event cannot have a probability less than 0 or more than 1.
• The sum of probabilities for all possible outcomes must equal 1, ensuring that one of the outcomes will certainly occur.

These rules ensure that probabilities are logical and consistent, reflecting the real-world expectations of the events they describe. If a given probability model does not meet these rules, it suggests that there is an error in the model, making it unreliable for predicting outcomes.

Die Roll Probability

Roll a classic 6-sided die, and you'll find that each side is equally likely to appear. This means each face of the die should have a probability of appearing of \( \frac{1}{6} \). In the exercise, we review whether the probabilities meet the rule where all outcomes must sum to 1.
Using given probabilities \( P(1) = 0, P(2) = \frac{1}{6}, P(3) = \frac{1}{3}, P(4) = \frac{1}{3}, P(5) = \frac{1}{6}, P(6) = 0 \), we confirm:

• The sum of these probabilities is 1.
• All the individual probabilities are between 0 and 1.

Even though numbers on the dice appear unequally likely, they satisfy the basic rules for probability. Hence, this setup is legitimate.

Student Enrollment Probability

When analyzing student enrollment, probability can help quantify different enrollment scenarios. The goal is to have probabilities that reflect realistic enrollment patterns.
In this case, we are given probabilities for four types of student enrollment: full-time and part-time for both genders. However, these probabilities are problematic because their sum is 1.41, exceeding 1.
A legitimate probability model should sum to exactly 1. This error suggests that we've overestimated the likelihood of these enrollment statuses or counted scenarios multiple times. It's essential to reassess and adjust these probabilities to meet the basic addition rule of probabilities.

Card Deck Probability

The standard deck of cards contains 52 cards, divided equally among four suits: clubs, diamonds, hearts, and spades. Each suit should ideally have the same probability of being drawn.
According to the exercise, each suit is given slightly skewed probabilities, but they do sum to 1: \( \frac{12}{52}, \frac{12}{52}, \frac{12}{52}, \frac{16}{52} \). However, each suit should have an equal probability of \( \frac{13}{52} \) since each suit has 13 cards.
Thus, while the sum is correct, the distribution is not. Each suit must be equally probable in a fair deck, meaning adjustments are necessary to reflect accurate probabilities that align with the real composition of a deck of cards.