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Chapter 2: Problem 16

An individual is presented with three different glasses of cola, labeled \(C, D\), and \(P\). He is asked to taste all three and then list them in order of preference. Suppose the same cola has actually been put into all three glasses. a. What are the simple events in this ranking experiment, and what probability would you assign to each one? b. What is the probability that \(C\) is ranked first? c. What is the probability that \(C\) is ranked first and \(D\) is ranked last?

Short Answer

Expert verified
a: 6 events, each with probability \( \frac{1}{6} \); b: \( \frac{1}{3} \); c: \( \frac{1}{6} \).

Step by step solution

01

Identify Simple Events

Since the cola in each glass is identical, any ranking is equally likely. The possible rankings are permutations of the labels \( C, D, P \). Hence, the simple events are: \( (C, D, P), (C, P, D), (D, C, P), (D, P, C), (P, C, D), (P, D, C) \).
02

Assign Probabilities to Simple Events

Since all the colas are identical, each of the 6 rankings is equally probable. With a total of 6 possible permutations, each event thus has a probability of \( \frac{1}{6} \).
03

Probability of C Being Ranked First

For \( C \) to be ranked first, the possible rankings are: \( (C, D, P) \) and \( (C, P, D) \). There are 2 such rankings. Thus the probability that \( C \) is ranked first is \( \frac{2}{6} = \frac{1}{3} \).
04

Probability of C Ranked First and D Ranked Last

For \( C \) to be ranked first and \( D \) last, the only suitable ranking is \( (C, P, D) \). There is 1 such ranking. So, the probability is \( \frac{1}{6} \).

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Key Concepts

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

Permutations
Permutations are arrangements where the order of items matters. Consider entries labeled A, B, and C. Arranging these labels in different ways is what permutations are about. Order is key here. For the glasses of cola placed in front of an individual, each unique sequence such as (C, D, P) or (P, D, C) represents a distinct permutation. Permutations consider all possible configurations:
  • The order of tasting or ranking matters.
  • Each unique sequence is a different permutation.
The number of permutations can be determined using factorials. If there are three items to arrange, the number of permutations is 3! (3 factorial), which equals 6. These permutations form the foundation of scenarios that can happen in the ranking experiment.
Ranking Experiment
In a ranking experiment, participants list items based on preferences. The task may involve comparing all items against each other and then determining a preferred order. Here鈥檚 how the ranking experiment works:
  • Identify all items to be ranked. In this case, they are glasses of cola labeled as C, D, and P.
  • The participant samples each item and chooses a preferred sequence or order.
  • Each possible order of the items is an outcome of the ranking experiment.
What makes this interesting in probability theory is that every arranging or ranking of the glasses results in a different outcome or simple event. In this exercise, the primary aim is to determine these simple events and analyze them.
Simple Events
Simple events are the individual outcomes in a probability experiment. In the context of ranking experiments, a simple event is a specific order in which items are arranged based on preference. Each permutation represents a simple event:
  • (C, D, P)
  • (C, P, D)
  • (D, C, P)
  • (D, P, C)
  • (P, C, D)
  • (P, D, C)
These represent all possible unique sequences for ranking the three cola glasses. Since the colas are identical, these events are the same in terms of preference and each event is "simple" because it cannot be broken down into more basic events.
Equally Likely Outcomes
In contexts where there is no bias or distinguishing factor between outcomes, events can be assumed to be equally likely. Here, since every glass contains the same cola, the outcomes of any ranking are assumed to be equally probable.Key points to consider:
  • Every permutation of items has the same chance of occurring.
  • For six possible outcomes in our example of cola ranking, each has a probability of \(\frac{1}{6}\).
  • Equal likelihood simplifies calculation of probabilities across events.
This equal probability stems from the uniformity in conditions, where no single outcome has an advantage over another, making calculations in probability straightforward and fair.

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Most popular questions from this chapter

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