91Ó°ÊÓ

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk. $$ \begin{array}{lccc} \hline & \text { Small } & \text { Medium } & \text { Large } \\ \hline \text { Regular } & 14 \% & 20 \% & 26 \% \\ \text { Decaf } & 20 \% & 10 \% & 10 \% \\ \hline \end{array} $$ Consider randomly selecting such a coffee purchaser. a. What is the probability that the individual purchased a small cup? A cup of decaf coffee? b. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability? c. If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

Step by step solution

01

Understanding the Problem

We have a table that presents the probability distribution of different types of coffee being purchased at an airport kiosk based on size and type. We need to calculate probability in various contexts: probability of buying a small cup, a decaf cup, a conditional probability if the cup size or type is known, and compare probabilities.

02

Calculating Probability of a Small Cup

To find the probability of purchasing a small cup, we sum the probabilities associated with the small size: \( P( ext{Small}) = P( ext{Small, Regular}) + P( ext{Small, Decaf}) = 0.14 + 0.20 = 0.34 \).

03

Calculating Probability of Decaf Coffee

To find the probability that the coffee purchased is decaf, add probabilities for each decaf size: \( P( ext{Decaf}) = P( ext{Small, Decaf}) + P( ext{Medium, Decaf}) + P( ext{Large, Decaf}) = 0.20 + 0.10 + 0.10 = 0.40 \).

04

Conditional Probability of Decaf given Small Cup

The notation \( P( ext{Decaf} | ext{Small}) \) represents the conditional probability of having decaf coffee given a small cup is chosen. Using the formula: \( P( ext{Decaf} | ext{Small}) = \frac{P( ext{Small, Decaf})}{P( ext{Small})} = \frac{0.20}{0.34} \approx 0.588 \). This means about 58.8% of those who choose a small cup opt for decaf coffee.

05

Conditional Probability of Small given Decaf

We want to find \( P( ext{Small} | ext{Decaf}) \), the probability of a small size being chosen given decaf coffee. Using the formula: \( P( ext{Small} | ext{Decaf}) = \frac{P( ext{Small, Decaf})}{P( ext{Decaf})} = \frac{0.20}{0.40} = 0.50 \). This indicates that 50% of the decaf purchases are small cups.

06

Comparison and Interpretation

The probability of choosing a small cup without conditions (unconditional) is 34%, while the conditional probability given decaf is 50%. This indicates that knowing the coffee is decaf increases the chance that it is a small cup, compared to the unconditional probability.

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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 statistical concept that shows how probabilities are spread across different outcomes. In the context of our coffee purchasing scenario, the probability distribution is presented in a table displaying different types of coffee choices marked by size (small, medium, large) and type (regular, decaf). Each entry in this table represents the probability of a specific coffee type and size combination being selected at random.

• The table entries indicate a probability like 14% for small regular coffee.
• These percentages total 100%, covering all possible coffee purchase options.
• Understanding the distribution helps analysts determine the likelihood of specific coffee options being chosen.

This distribution helps us easily calculate unconditional and conditional probabilities, which are essential for making predictions in scenarios where choices are based on multiple factors.

Unconditional Probability

Unconditional probability refers to the likelihood of an event occurring without any other influencing conditions. It is akin to pondering the odds of something happening in a complete vacuum of context.
In our exercise, two types of unconditional probabilities were calculated:

• Probability of purchasing a small cup, achieved by summing probabilities of all small-sized options: \( P(\text{Small}) = 0.34 \), or 34%.
• Probability of purchasing a decaf cup: \( P(\text{Decaf}) = 0.40 \), which is 40%.

These calculations are crucial because they provide a baseline, showing the general tendency of an event without additional conditions or filters applied.

Probability Interpretation

Probability interpretation involves understanding the meaning and implications of probability results, particularly conditional probabilities, in our exercise. Conditional probabilities offer insights under specific conditions, differing from unconditional ones that lack such context.
For instance, the conditional probability \( P(\text{Decaf} | \text{Small}) \), the probability of a coffee being decaf given it's already known to be a small cup, was calculated to be approximately 58.8%.

• This indicates a higher likelihood of choosing decaf when the selection is specifically within small-sized coffees, compared to the unconditional 40% probability.
• Similarly, \( P(\text{Small} | \text{Decaf}) \) found that 50% of decaf purchases were small cups, noticeably higher than the outright 34% probability for small cups.

Understanding conditional probabilities not only refines our expectations but also highlights how new information can alter prior assumptions, showcasing the dynamic interplay between different variables in probability.