Problem 10 - Basic Statistics for Business and Economics with Student

Question

In a binomial situation \(n=5\) and \(\pi=.40 .\) Determine the probabilities of the following events using the binomial formula. a. \(x=1\) b. \(x=2\)

Step-by-Step Solution

Answer

a. 0.2592 b. 0.3456

Step 1: Identify the Binomial Formula

The binomial probability formula is \( P(X = x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \). Here, \( n \) is the number of trials, \( x \) is the number of successful trials, and \( \pi \) is the probability of success on a single trial.

Step 2: Calculate \(P(x=1)\)

First, identify the known values for \(x=1\): \(n=5\), \(x=1\), and \(\pi=0.4\). Then substitute these values into the binomial formula: \( P(X = 1) = \binom{5}{1} (0.4)^1 (0.6)^{5-1} \). Calculate \(\binom{5}{1} = 5\), then \((0.4)^1 = 0.4\), and \((0.6)^4 = 0.1296\). Thus, \( P(X = 1) = 5 \times 0.4 \times 0.1296 = 0.2592 \).

Step 3: Calculate \(P(x=2)\)

Next, identify the known values for \(x=2\): \(n=5\), \(x=2\), and \(\pi=0.4\). Substituting these values into the binomial formula gives: \( P(X = 2) = \binom{5}{2} (0.4)^2 (0.6)^{5-2} \). Calculate \(\binom{5}{2} = 10\), then \((0.4)^2 = 0.16\), and \((0.6)^3 = 0.216\). Thus, \( P(X = 2) = 10 \times 0.16 \times 0.216 = 0.3456 \).

Key Concepts

Binomial Distribution

The binomial distribution is a key concept in probability theory. It models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. For example, flipping a coin several times and counting the number of heads gives you a simple binomial distribution if each flip has the same chance, say 50%, of landing heads.
The binomial distribution is characterized by two parameters:

• \( n \): the number of trials, representing how many times the experiment is conducted.
• \( \pi \): the probability of success for a single trial.

Using these parameters, the number of successful outcomes (like getting heads in our coin flip example) follows the binomial probability formula:

• \( P(X = x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \)

This formula helps in calculating the probability of achieving exactly \( x \) successes in \( n \) trials.

Probability Theory

Probability theory is the mathematical framework used to study and understand random phenomena. It helps calculate how likely an event is to happen. When working with probability, we often consider all possible outcomes, which are called a sample space.
When thinking about probability:

• Each event has a probability between 0 (impossible) and 1 (certain).
• The sum of probabilities of all possible outcomes of a trial is always 1.

In probability theory, events can be independent (the outcome of one event doesn't affect another) or dependent (outcomes affect each other).
With binomial distribution problems, you often deal with independent events, such as a series of trials where the probability stays consistent across each trial.

• For instance, in the example of calculating probabilities using the binomial formula, each flip of the coin, draw from a deck, or instance of success or failure is treated as an independent event with a consistent probability connected to the event.

Statistical Calculations

Statistical calculations are used to find out meaningful insights from data. For binomial probabilities, statistical calculations involve using combinations and probability to determine the likelihood of a specific number of successes in repeated trials.
Here's a breakdown of the important steps in these calculations:

• Combinations: \( \binom{n}{x} \) represents the number of ways to choose \( x \) successes from \( n \) trials, which is calculated as \( \frac{n!}{x!(n-x)!} \). This step reflects different possible arrangements of successes in trials.
• Power Calculation: Next, you calculate the powers of the probability of success and failure. \( \pi^x \) and \((1-\pi)^{n-x}\) represent these calculations.
• Final Probability: By multiplying \( \binom{n}{x} \), \( \pi^x \), and \((1-\pi)^{n-x}\), you arrive at the final probability of achieving exactly \( x \) successes.

An understanding of statistical calculations is crucial for making sense of real-world data, especially when evaluating results using probability theories like the binomial distribution. By mastering these steps and principles, you can effectively interpret statistical outcomes and act based on calculated probabilities.