91Ó°ÊÓ

Berdine's Chicken Factory has several stores in the Hilton Head, South Carolina, area. When interviewing applicants for server positions, the owner would like to include information on the amount of tip a server can expect to earn per check (or bill). $$ \begin{array}{|ccc|} \hline \text { Amount of Tip } & \text { Number } \\ \hline \$ 0 \text { up to } \$ 20 & 200 \\ 20 \text { up to } & 50 & 100 \\ 50 \text { up to } & 100 & 75 \\ 100 \text { up to } & 200 & 75 \\ 200 \text { or more } & \frac{50}{500} \\ \text { Total } & & 500 \\ \hline \end{array} $$ a. What is the probability of a tip of \(\$ 200\) or more? b. Are the categories "\$0 up to \$20," "\$20 up to \$50," and so on considered mutually exclusive? c. If the probabilities associated with each outcome were totaled, what would that total be? d. What is the probability of a tip of up to \(\$ 50 ?\) e. What is the probability of a tip of less than \(\$ 200 ?\)

Step by step solution

01

Calculate Probability for $200 or more

To find the probability of receiving a tip of \(200 or more, divide the number of such tips by the total number of tips. The number of tips \)200 or more is 50 and the total number of tips is 500. Thus, the probability is calculated as follows:\[ P( ext{tip} \geq 200) = \frac{50}{500} = 0.10 \]

02

Check for Mutually Exclusive Categories

Categories are considered mutually exclusive if no outcome can belong to more than one category. In this tip table, each amount range is distinct and non-overlapping, ensuring that a tip can only fall into one category. Thus, these categories are mutually exclusive.

03

Sum Probabilities for All Categories

The probability associated with each outcome must sum to 1 for a complete probability distribution. Calculate the probability for each category and add them up:\[ \begin{align*}P( ext{tip} < 20) &= \frac{200}{500} = 0.40, \P(20 \leq ext{tip} < 50) &= \frac{100}{500} = 0.20, \P(50 \leq ext{tip} < 100) &= \frac{75}{500} = 0.15, \P(100 \leq ext{tip} < 200) &= \frac{75}{500} = 0.15, \P(200 \leq ext{tip}) &= 0.10 \end{align*} \]Adding these probabilities gives us:\[ 0.40 + 0.20 + 0.15 + 0.15 + 0.10 = 1.00 \]

04

Calculate Probability for Tip Up to $50

To find the probability of receiving a tip up to \(50, add together the probabilities of the relevant categories: less than \)20 and \(20 to \)50:\[ P( ext{tip} < 50) = P( ext{tip} < 20) + P(20 \leq ext{tip} < 50) = 0.40 + 0.20 = 0.60 \]

05

Calculate Probability for Tip Less Than $200

For the probability of receiving a tip less than \(200, sum the probabilities of all categories below \)200:\[\begin{align*}P( ext{tip} < 200) &= P( ext{tip} < 20) + P(20 \leq ext{tip} < 50) + P(50 \leq ext{tip} < 100) + P(100 \leq ext{tip} < 200) \&= 0.40 + 0.20 + 0.15 + 0.15 = 0.90\end{align*}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Mutually Exclusive Events

In probability, mutually exclusive events are those which cannot happen simultaneously. Think of rolling a dice; you cannot roll a three and a five at the same time—it’s one or the other. In this exercise at Berdine's Chicken Factory, each tip amount category represents a set of mutually exclusive events. Why? Because a tip of exactly $45, for instance, can only fall into the "$20 up to $50" category, and not into any other.

• Mutually exclusive means distinct and non-overlapping categories.
• In our exercise, each tip category is exclusive; a specific tip can't belong to two categories.

Understanding this concept ensures that when calculating probabilities, we don’t count the same event multiple times, which could lead to incorrect total probabilities.

Probability Distribution

A probability distribution tells us how a total probability of 1 is distributed across different events. It's like cutting a pie, where the whole pie equals the probability of all events happening, summing up to 1.
For the tip amounts at Berdine's, we distribute the total probability across each category, allowing the owner to communicate expected earnings accurately to applicants.

• Each category's probability is calculated as the number of occurrences divided by the total occurrences.
• All these probabilities should add up to 1 to make a complete distribution.

This distribution is essential because it provides a comprehensive picture of chances for different outcomes, assisting the owner in predicting the earning range for servers.

Probability Calculation

Calculating probability involves finding out how likely an event is to happen. In our tips example, this means determining how often a given tip category appears. To calculate, simply divide the number of favorable outcomes by the total number of outcomes.

• For example, the probability of getting a tip of \(200 or more is calculated as \( \frac{50}{500} = 0.10 \).
• Similarly, for tips of less than \)50, combine the outcomes for tips "up to \(20" and "\)20 up to $50."

These calculations help us grasp the likely financial returns and plan better. Being able to easily compute these likelihoods allows prospective employees to have a realistic expectation of their tips.

Step-by-Step Solutions

Working through a problem with a step-by-step approach breaks down complex processes into manageable pieces, enhancing understanding. Let's revisit our exercise using this method.
Begin with stating what you need, such as the probability of tips over $200. Identify the step to achieve it—the successful tips divided by total tips. Continue systematically till you get your desired answer. Is each tip category exclusive? Check! Add them up—do they total to 1? Yes! Each calculation or validation step done sequentially confirms our understanding.

• Breaking down ensures you don't miss any category or calculation.
• Chronological steps offer clarity in learning and verification.

This approach not only simplifies complex problems but instills confidence by reinforcing that each part of the solution is correct and contributes to the final answer.