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. 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 upface: $$ \begin{array}{lll} P(1)=0 & P(2)=1 / 6 & P(3)=1 / 3 \\ P(4)=1 / 3 & P(5)=1 / 6 & P(6)=0 \end{array} $$ b. Deal a card from a shuffled deck: $$ \begin{array}{rlrl} P(\text { clubs }) & =12 / 52 & P(\text { diamonds }) & =12 / 52 \\ P(\text { hearts }) & =12 / 52 & P(\text { spades }) & =16 / 52 \end{array} $$ c. Choose a college student at random and record sex and enrollment status: $$ \begin{array}{rlrl} P(\text { female full-time }) & =0.56 & P(\text { male full-time }) & =0.44 \\\ P(\text { female part-time }) & =0.24 & P(\text { male part-time }) & =0.17 \end{array} $$

Step by step solution

01

Verify Probability Rules for Each Case

Recall the two basic rules of probability: 1) The probability of each outcome must be between 0 and 1, inclusive. 2) The sum of probabilities for all possible outcomes in the sample space must equal 1.

02

Evaluate Legitimacy of Case (a)

For rolling a six-sided die, probabilities assigned are: \( P(1)=0, P(2)=1/6, P(3)=1/3, P(4)=1/3, P(5)=1/6, P(6)=0 \). Verify they are between 0 and 1. All values comply. Now, sum the probabilities: \( 0 + 1/6 + 1/3 + 1/3 + 1/6 + 0 = 1 \). Therefore, case (a) is legitimate.

03

Evaluate Legitimacy of Case (b)

The probabilities for dealing a card are: \( P(\text{clubs}) = 12/52, P(\text{diamonds}) = 12/52, P(\text{hearts}) = 12/52, P(\text{spades}) = 16/52 \). Each is between 0 and 1. Sum them: \( 12/52 + 12/52 + 12/52 + 16/52 = 52/52 = 1 \). Thus, case (b) is legitimate.

04

Evaluate Legitimacy of Case (c)

Probabilities for choosing a college student are: \( P(\text{female full-time}) = 0.56, P(\text{male full-time}) = 0.44, P(\text{female part-time}) = 0.24, P(\text{male part-time}) = 0.17 \). Check each value is between 0 and 1. Sum them: \( 0.56 + 0.44 + 0.24 + 0.17 = 1.41 \). Since 1.41 > 1, case (c) is not legitimate.

05

Conclusion

Case (a) and case (b) are legitimate as they satisfy all rules of probability. However, case (c) is not legitimate as the sum of probabilities exceeds 1.

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Key Concepts

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

Probability Rules

Probability rules are basic principles that govern the way probabilities are assigned to events. There are two critical rules to remember:

• The probability of any specific outcome must always be between 0 and 1, inclusive. This means it can be zero (event will not happen) or one (event will certainly happen), but cannot exceed these bounds.
• The total sum of probabilities for all possible outcomes in a sample space must equal 1. This ensures that all possible events are accounted for.

Understanding these rules is fundamental as they form the basis for determining whether a probability model is legitimate or not. If any of these rules are violated, the model is considered incorrect or illegitimate.
Adhering to these rules is crucial for ensuring that probability calculations yield reliable predictions.

Sample Space

The sample space is the set of all possible outcomes in a probability experiment. It is often denoted by the symbol \( S \). Determining the correct sample space is vital as it defines the frame in which probabilities must add up to 1.

For example, when rolling a six-sided die, the sample space is \( S = \{1, 2, 3, 4, 5, 6\} \). This means any outcome of the die roll must be one of these six numbers. Each outcome is a distinct element of \( S \), and together they encompass every possible result of that experiment.

Similarly, when drawing a card from a deck, the sample space consists of all 52 cards. This ensures you are considering the entire deck when calculating the probability for any specific suit or card. Properly defining the sample space is fundamental to the accurate assignment of probabilities.

Probability Assignment

Probability assignment is the process of assigning a numerical value (between 0 and 1) to each possible outcome in the sample space. This assignment quantifies how likely any particular event is to happen.

For instance, in a fair six-sided die, each side has an equal probability of landing face up. This is calculated as \( P(x) = \frac{1}{6} \) for each side, since there are 6 equal possibilities.

Another example is dealing cards from a standard deck. Each suit—hearts, diamonds, clubs, and spades—must have an assigned probability based on the number of cards per suit. Correct probability assignment ensures that these probabilities sum up correctly within the rules of the sample space, reinforcing the model's legitimacy.

In sound probability assignments, all probabilities collectively spell out the likelihood of every scenario under the defined conditions of the experiment.

Legitimacy of Probability Models

For a probability model to be considered legitimate, it must satisfy the probability rules laid out earlier. This involves ensuring the following:

• Each assigned probability lies within 0 and 1.
• The sum of probabilities across the whole sample space equals exactly 1.

Let's examine a simple example using a deck of cards. Assuming correctness in counts, the assigned probabilities seem correctly totaled, yet the distribution must reflect reality based on constraints, like the fixed number of cards per suit.

In some cases, probabilities might mistakenly sum to a number greater than 1. In such cases, the model is deemed illegitimate, as seen in the situation of selecting college students, where the total probabilities exceeded 1. This violates probability rules, indicating potential errors in the probability assignment or sample space definition.

Ensuring the legitimacy of probability models is crucial to modeling real-world uncertainties accurately and reliably.