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Chapter 12: Problem 21

Winning the ACC Toumament. The annual Atlantic Coast Conference men's basketball tournament has temporarily taken Joe's mind off of the Cleveland Indians. He says to himself, "I think that Notre Dame has probability \(0.05\) of winning. North Carolina's probability is twice Notre Dame's, and Duke's probability is four times Notre Dame's." (a) What are Joe's personal probabilities for North Carolina and Duke? (b) What is Joe's personal probability that 1 of the 12 teams other than Notre Dame, North Carolina, and Duke will win the tourmament?

Short Answer

Expert verified
(a) North Carolina: 0.10, Duke: 0.20; (b) Other 12 teams: 0.65.

Step by step solution

01

Understand Notre Dame's Probability

Notre Dame has a probability of winning the tournament given as 0.05. This will be used as the base probability to calculate the probabilities for North Carolina and Duke.
02

Calculate North Carolina's Probability

The problem states that North Carolina's probability is twice Notre Dame's. Thus, we compute: \( P(\text{North Carolina}) = 2 \times 0.05 = 0.10 \).
03

Calculate Duke's Probability

Similarly, Duke's probability is four times Notre Dame's probability, so: \( P(\text{Duke}) = 4 \times 0.05 = 0.20 \).
04

Calculate Probability for Other Teams

Since probabilties must add up to 1, we calculate the total probability of Notre Dame, North Carolina, and Duke: \( 0.05 + 0.10 + 0.20 = 0.35 \). The probability that one of the other 12 teams wins is: \( 1 - 0.35 = 0.65 \).

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Key Concepts

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

Personal Probability
Personal probability is a subjective measure of the likelihood of an event happening. Unlike the classical or frequentist probability, which is based on the frequency of events occurring in the long run, personal probability depends on an individual's judgment or belief. Here's how you might think about it:
  • *Subjective*: Often based on personal opinion, feelings, or intuition.
  • *Contextual*: Can vary from person to person, as it's influenced by one's own experiences and information at hand.
  • *Non-empirical*: May not rely solely on mathematical calculations or past data.
In Joe's situation, his personal probabilities for Notre Dame, North Carolina, and Duke are based solely on his subjective belief about the chances of each team winning. This means that while he believes Notre Dame has a 5% chance of winning based on his judgment, other people might assign different probabilities based on their perspectives.
Probability Calculation
Calculating probability often involves using given information to find out the chances of different outcomes. In the case of personal probability, as discussed in Joe's scenario, we have to rely on what is provided and derive the rest. We used the following simple arithmetic principles for this:
  • North Carolina's probability is calculated as twice the probability of Notre Dame: \[ P(\text{North Carolina}) = 2 \times P(\text{Notre Dame}) = 2 \times 0.05 = 0.10 \]
  • Duke's probability is four times Notre Dame's probability: \[ P(\text{Duke}) = 4 \times P(\text{Notre Dame}) = 4 \times 0.05 = 0.20 \]
The calculated probabilities for each team give insight into Joe's subjective view of their chances. This process highlights how we can scale and adjust probabilities to reflect different scenarios and personal beliefs.
Probability Addition Rule
The probability addition rule is crucial when dealing with non-mutually exclusive events. However, in our scenario with Joe, we deal with mutually exclusive events, since one team winning precludes others from winning. Key Steps:
  • First, add up the probabilities for Notre Dame, North Carolina, and Duke:
    • \[ P(\text{Notre Dame}) + P(\text{North Carolina}) + P(\text{Duke}) = 0.05 + 0.10 + 0.20 = 0.35 \]
  • The probability that any one of the remaining 12 teams wins is the complement of these combined probabilities. Use the complement rule:\[ P(\text{Other Teams}) = 1 - P(\text{Notre Dame, North Carolina, Duke}) = 1 - 0.35 = 0.65 \]
So, this rule allows us to calculate the likelihood of any other event occurring in a scenario, enhancing our understanding of the entire probability space.

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Most popular questions from this chapter

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