Problem 62 - Basic Statistics for Business and Economics with Student

Question

An Internet company located in Southern California has season tickets to the Los Angeles Lakers basketbal games. The company president always invites one of the four vice presidents to attend games with him, and claims he selects the person to attend at random. One of the four vice presidents has not been invited to attend any of the last five Lakers home games. What is the likelihood this could be due to chance?

Step-by-Step Solution

Answer

The likelihood is approximately 23.7%.

Step 1: Understand the Situation

There are four vice presidents and the president selects one person randomly for each game. Each vice president has an equal probability of being chosen, which is \( \frac{1}{4} \). The questioned scenario is that one specific vice president was not chosen for any of the last five games.

Step 2: Calculate the Probability for Each Individual Game

The probability that a specific vice president is not chosen for a single game is \( 1 - \frac{1}{4} = \frac{3}{4} \).

Step 3: Compute the Probability for Five Consecutive Games

We want to find the probability that a specific vice president is not selected for five games in a row. This is a situation involving repeated independent events. The probability over these consecutive games is \[\left( \frac{3}{4} \right)^5 = \frac{243}{1024}.\]

Step 4: Interpret the Results

The probability \( \frac{243}{1024} \approx 0.237 \) means that there is approximately a 23.7% chance of a specific vice president not being selected for any of the five games by pure chance.

Key Concepts

Random Selection in Probability Theory

Random selection is an essential concept in probability theory. It reflects how choices are made without preference or pattern. For example, when the company president chooses a vice president to attend a game, he claims it's done randomly. This means each vice president should have an equal likelihood of being selected.
In this case, with four vice presidents, each has a \( \frac{1}{4} \) chance of being chosen for any game. Random selection ensures fairness and equality, allowing us to apply probability calculations. Understanding randomness helps us analyze the likelihood of certain outcomes happening purely by chance.

Understanding Independent Events

Independent events are a fundamental concept in probability. Two events are independent if the outcome of one does not influence the outcome of the other. In our scenario, selecting a vice president for one game does not affect the selections for the other games.
For example, the probability of not selecting a specific vice president for a single game is \( \frac{3}{4} \). This probability remains the same for each of the five games, as each game's selection does not depend on the others. Recognizing events as independent helps us simplify complex probability problems, enabling straightforward calculations. This assumption is crucial in calculating probabilities over consecutive events.

Calculating Probabilities of Repeated Events

Calculating probabilities involves determining the likelihood of specific outcomes. For repeated independent events, like the repeated selection of vice presidents, we use the multiplication rule for probabilities.
To find the probability that a specific vice president is not selected for five consecutive games, we multiply the probability of not being selected in one game across all five games: \[\left( \frac{3}{4} \right)^5 = \frac{243}{1024}\]This results in a probability of approximately 23.7%. This calculation shows that despite each game being independent, the combination of events can lead to a lower likelihood of repetitive exclusion.

Game Theory and Strategic Decisions

Game theory provides a framework for analyzing situations where multiple decision-makers interact strategically. It can apply to scenarios like our exercise, where decisions might not seem strategic initially but have underlying consequences.
In this example, if the president's selection were influenced by unseen factors, it could deviate from pure random selection, suggesting a strategic element. However, assuming random selection suggests there is no strategy. Game theory helps us explore possibilities beyond surface-level randomness, providing insights into decisions and ensuring fairness. Understanding this can highlight potential biases or strategic influences in decision-making processes.