Problem 19 - Basic Statistics for Business & Economics

Question

In a binomial distribution, \(n=8\) and \(\pi=.30 .\) Find the probabilities of the following events. a. \(x=2\) b. \(x \leq 2\) (the probability that \(x\) is equal to or less than 2 ). c. \(x \geq 3\) (the probability that \(x\) is equal to or greater than 3 ).

Step-by-Step Solution

Answer

a) 0.2541, b) 0.5094, c) 0.4906

Step 1: Identify Parameters

The exercise specifies a binomial distribution where \( n = 8 \) and \( \pi = 0.30 \). Here, \( n \) represents the number of trials and \( \pi \) represents the probability of success in each individual trial.

Step 2: Find the Probability for \( x=2 \)

The probability of \( x \) successes in a binomial distribution is given by the formula: \[ P(x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \] For \( x = 2 \):\[ P(2) = \binom{8}{2} (0.30)^2 (0.70)^6 \] Computing this gives:\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]\[ (0.30)^2 = 0.09 \]\[ (0.70)^6 = 0.117649 \] Thus,\[ P(2) = 28 \times 0.09 \times 0.117649 \approx 0.2541 \]

Step 3: Find the Probability for \( x \leq 2 \)

The probability that \( x \leq 2 \) is the sum of the probabilities of \( x = 0 \), \( x = 1 \), and \( x = 2 \). Using the binomial formula:\[ P(0) = \binom{8}{0} (0.30)^0 (0.70)^8 = 1 \times 1 \times 0.05764801 \approx 0.0576 \]\[ P(1) = \binom{8}{1} (0.30)^1 (0.70)^7 = 8 \times 0.3 \times 0.0823543 \approx 0.1977 \]Combining these probabilities:\[ P(x \leq 2) = P(0) + P(1) + P(2) \approx 0.0576 + 0.1977 + 0.2541 \approx 0.5094 \]

Step 4: Find the Probability for \( x \geq 3 \)

The probability that \( x \geq 3 \) can be found by subtracting \( P(x \leq 2) \) from 1:\[ P(x \geq 3) = 1 - P(x \leq 2) \approx 1 - 0.5094 = 0.4906 \]

Key Concepts

Probability Calculation

Probability calculation using a binomial distribution involves determining the likelihood of a specific number of successes in a set number of trials. In binomial distribution, each trial is independent, with two possible outcomes: success or failure. Calculations are made using the given probability of success for each trial.
To calculate the probability of a specific number of successes, such as 2 successes out of 8 trials, you can apply the binomial formula. By reassessing the distribution, you can calculate probabilities for different outcomes, such as fewer than or at least a certain number of successes.
These calculations play a crucial role in hypotheses testing and real-life decision-making scenarios, where predicting outcomes based on probabilities helps in drawing evidence-based conclusions.

Binomial Formula

The binomial formula is a mathematical equation used to calculate the probability of obtaining a given number of successes in a set number of trials. This is particularly useful when dealing with binomial distributions. The formula is expressed as follows: \[ P(x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \]Where: - \( P(x) \) is the probability of getting \( x \) successes.- \( \binom{n}{x} \) represents the number of combinations of \( n \) items taken \( x \) at a time.
- \( \pi \) is the probability of success on a single trial.- \( (1-\pi) \) is the probability of failure. This formula allows you to calculate the probability for exactly \( x \) successes in \( n \) trials, as seen in the exercise when calculating \( P(x=2) \). It is a cornerstone for probability theory and is frequently used in statistical analyses.

Success Probability

In a binomial distribution, success probability, denoted as \( \pi \), is the likelihood that a single trial results in success. For each trial, \( \pi \) remains constant throughout the process, determining the probability of success regardless of previous outcomes.
For example, if \( \pi = 0.30 \), there's a 30% chance of success on any given trial. Understanding \( \pi \) is vital as it directly affects the outcome of the probability calculations and shapes the distribution. When calculating probabilities using the binomial formula, \( \pi \) plays a significant role in determining how likely different numbers of successes are, as it is used in the formula's component for success probability, \( \pi^x \).
The success probability helps in various applications, from quality control testing to making predictions in uncertain scenarios.

Number of Trials

The number of trials, denoted as \( n \), represents how many times an experiment is conducted in a binomial distribution scenario. Each trial should be independent, meaning the outcome of one trial does not affect another.
In the example provided, \( n = 8 \) indicates that the experiment, or the process of checking for success, is carried out 8 times. Calculating the probability of different events, such as exactly 2 successes, or success in less than or more than a certain number of trials, relies on \( n \).
It is essential to accurately determine \( n \) in probability equations, as changes in \( n \) can significantly impact the final probability outcomes.

• The number of trials affects the height and shape of the probability mass function.
• Understanding \( n \) helps determine expected values and variances.

Given its importance, \( n \) is always a central parameter in binomial distribution problems.