Problem 40 - Basic Statistics for Business & Economics

Question

Tire and Auto Supply is considering a 2 -for- 1 stock split. Before the transaction is finalized, at least two-thirds of the 1,200 company stockholders must approve the proposal. To evaluate the likelihood the proposal will be approved, the CFO selected a sample of 18 stockholders. He contacted each and found 14 approved of the proposed split. What is the likelihood of this event, assuming two-thirds of the stockholders approve?

Step-by-Step Solution

Answer

The likelihood that exactly 14 out of 18 stockholders approve is calculated using the binomial probability formula.

Step 1: Understand the Problem

We need to calculate the likelihood that exactly 14 out of 18 stockholders approve the stock split, assuming that each has a probability of 2/3 of approving.

Step 2: Identify the Distribution

The problem involves determining probabilities related to a number of approvals, which is a typical application of the binomial distribution. Define the number of trials as 18 (the number of stockholders contacted) and the number of successes as 14.

Step 3: Set the Parameters

In a binomial distribution, the probability of success \( p = \frac{2}{3} \). The number of trials \( n = 18 \), and we seek the probability of \( k = 14 \) approvals.

Step 4: Write the Binomial Probability Formula

The probability of exactly \( k \) successes in \( n \) trials is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( \binom{n}{k} \) is the binomial coefficient.

Step 5: Calculate the Binomial Coefficient

The binomial coefficient \( \binom{18}{14} \) is calculated as:\[ \binom{18}{14} = \frac{18!}{14!(18-14)!} = \frac{18!}{14!4!} \]

Step 6: Calculate Probability Term p^k

Compute \( (\frac{2}{3})^{14} \). This represents the probability of 14 approvals.

Step 7: Calculate Probability Term (1-p)^(n-k)

Compute \( (\frac{1}{3})^{4} \). This represents the probability of 4 non-approvals.

Step 8: Calculate Exact Probability

Combine these calculations to find \( P(X = 14) \):\[ P(X = 14) = \binom{18}{14} \left(\frac{2}{3}\right)^{14} \left(\frac{1}{3}\right)^4 \]

Step 9: Compute the Numeric Result

Calculate the numeric value of \( P(X = 14) \) using a calculator or statistical software.

Key Concepts

Probability of Success

In a binomial distribution, the probability of success is crucial. It represents the likelihood of one specific outcome occurring in a single trial. In the context of our exercise, this success is a stockholder approving the stock split. Each stockholder has a probability of success of \( \frac{2}{3} \). This means that, on average, 2 out of every 3 stockholders would approve the stock split. Understanding the probability of success \( p \) helps us determine the expected number of successes over multiple trials. This expectation sets the stage for further calculations in binomial distribution exercises.
The concept highlights how likely each individual event will meet our criteria for success. It's a foundational element when predicting outcomes in statistical scenarios.

Binomial Coefficient

The binomial coefficient is what makes the binomial distribution so versatile in predicting exact outcomes. It determines the number of ways a specific number of successes can occur among a given number of trials. In our example, it represents the number of ways 14 successes (approvals) can be achieved out of 18 trials (stockholders contacted). Mathematically, it's represented as \( \binom{n}{k} \) and calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here, \( n \) is the total number of trials (18), and \( k \) is the number of desired successes (14).

• \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).
• \( k! \) and \((n-k)! \) are the factorials of \( k \) and \((n-k) \) respectively.

In our step-by-step solution, \( \binom{18}{14} \) is computed to find the number of combinations for this scenario. It provides a critical multiplicative factor for calculating the probability of exact outcomes in our query.

Probability Calculation

Calculating the probability of a specific number of successes involves a few distinct steps. After identifying the probability of success and the binomial coefficient, we combine them to compute an exact probability. This involves the probability formula for exactly \( k \) successes in \( n \) trials:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]Each component plays a vital role:

• \( \binom{n}{k} \): The binomial coefficient, as calculated earlier.
• \( p^k \): The probability of success raised to the power \( k \), representing 14 approvals.
• \((1-p)^{n-k} \): The probability of failure raised to the power of the remaining trials, representing 4 non-approvals in our case.

By substituting our known values into this formula, we calculate the probability specifically for 14 approvals among 18 stockholders:\[P(X = 14) = \binom{18}{14} \left(\frac{2}{3}\right)^{14} \left(\frac{1}{3}\right)^{4}\]This comprehensive combination of these factors provides the precise likelihood of the given scenario.