Problem 13 - Basic Statistics for Business & Economics

Question

An American Society of Investors survey found \(30 \%\) of individual investors have used a discount broker. In a random sample of nine individuals, what is the probability: a. Exactly two of the sampled individuals have used a discount broker? b. Exactly four of them have used a discount broker? c. None of them has used a discount broker?

Step-by-Step Solution

Answer

a. Approximately 0.2668 b. Approximately 0.0595 c. Approximately 0.0403

Step 1: Understanding the Problem

We need to find the probability of a certain number of individuals using a discount broker, given that the probability of one individual using it is \( 0.3 \). This is a binomial probability problem with \( n = 9 \) and \( p = 0.3 \).

Step 2: Applying the Binomial Probability Formula

The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the total number of trials, \( k \) is the number of successes, and \( p \) is the probability of success on each trial.

Step 3: Calculating Probability for Part a

For part a, we calculate \( P(X = 2) \). Using the formula: \( P(X = 2) = \binom{9}{2} (0.3)^2 (0.7)^{7} \). Calculate \( \binom{9}{2} = 36 \), and plugging into the formula gives \( 36 \times (0.3)^2 \times (0.7)^7 \approx 0.2668 \).

Step 4: Calculating Probability for Part b

For part b, we calculate \( P(X = 4) \). Using the same formula: \( P(X = 4) = \binom{9}{4} (0.3)^4 (0.7)^{5} \). Calculate \( \binom{9}{4} = 126 \), and plugging into the formula gives \( 126 \times (0.3)^4 \times (0.7)^5 \approx 0.0595 \).

Step 5: Calculating Probability for Part c

For part c, we calculate \( P(X = 0) \). Using the formula: \( P(X = 0) = \binom{9}{0} (0.3)^0 (0.7)^{9} \). As \( \binom{9}{0} = 1 \), it simplifies to \( (0.7)^9 \approx 0.0403 \).

Step 6: Final Check and Interpretation

Ensure all calculations are performed accurately. The results are probabilities of exactly two, four, and zero investors using a discount broker in the sample.

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of events occurring. It provides a framework for quantifying uncertainty and enables us to make predictions about future occurrences. In our exercise, we are concerned with finding the probability of a specific number of individual investors using a discount broker out of a random sample.

Probability theory involves various concepts, including:

• **Random Variables**: Variables whose values result from random phenomena.
• **Events**: Outcomes or sets of outcomes from a random process.
• **Probability Distributions**: Mathematical functions that provide the likelihood of each possible value of a random variable.

In this exercise, a binomial distribution is used, which is a specific type of probability distribution that applies to binomial experiments. This forms the basis for listings events and calculations under certain conditions specified, such as the probability of using a discount broker.

Statistics

Statistics is a field that focuses on the collection, analysis, interpretation, presentation, and organization of data. Within this discipline, probability plays a crucial role, as it allows statisticians to make inferences about a larger population based on sample data.

In the context of our problem, statistics help us understand how we can use past data, such as the survey indicating that 30% of individual investors use a discount broker, to make predictions about a sample. This involves concepts like:

• **Descriptive Statistics**: Summarizing and describing the features of a dataset.
• **Inferential Statistics**: Making predictions or inferences about a population based on a sample.

Using statistics, we interpret the calculated probabilities to understand how likely it is for two, four, or none of the sampled individuals to use a discount broker, hence informing decisions or assumptions for future occurrences.

Binomial Distribution

The binomial distribution is a discrete probability distribution used when there are two possible outcomes in an experiment. It is often used to model the number of successes in a fixed number of trials in situations where each trial is independent of the others, such as in the provided exercise.

Key characteristics of a binomial distribution include:

• **Number of Trials (n)**: The fixed number of independent experiments, in this case, nine individual investors.
• **Probability of Success (p)**: The probability that a single trial is successful. Here, it is 30%, or 0.3, for using a discount broker.
• **Number of Successes (k)**: The specific number of successful trials we are interested in (e.g., exactly two, four, or none).
• **Success and Failure**: Each trial results in either success (using a broker) or failure (not using a broker).

The binomial distribution formula, \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), is applied to calculate probabilities by substituting the values of \( n, p, \) and \( k \). Understanding this helps in solving similar problems where decisions are discrete and binomial in nature.