Problem 9 - Basic Statistics for Business & Economics

Question

In a binomial situation, \(n=4\) and \(\pi=.25 .\) Determine the probabilities of the following events using the binomial formula. a. \(x=2\) b. \(x=3\)

Step-by-Step Solution

Answer

a. \( P(x=2) = 0.2109375 \), b. \( P(x=3) = 0.046875 \).

Step 1: Understand the binomial formula

The binomial probability formula is used to calculate the probability of obtaining a fixed number of successes in a certain number of trials. It is given as: \( P(x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \) where \( n \) is the number of trials, \( x \) is the number of successes, and \( \pi \) is the probability of success on a single trial.

Step 2: Calculate probability for x=2

Substitute \( n=4 \), \( \pi=0.25 \), and \( x=2 \) into the binomial probability formula: \[ P(2) = \binom{4}{2} (0.25)^2 (0.75)^2 \]. First, calculate \( \binom{4}{2} = \frac{4!}{2!(4-2)!} = 6 \). Then, calculate \( (0.25)^2 = 0.0625 \) and \( (0.75)^2 = 0.5625 \). Multiply these results together: \( P(2) = 6 \times 0.0625 \times 0.5625 = 0.2109375 \).

Step 3: Calculate probability for x=3

Using the same approach, substitute \( n=4 \), \( \pi=0.25 \), and \( x=3 \) into the binomial formula: \[ P(3) = \binom{4}{3} (0.25)^3 (0.75)^1 \]. Calculate \( \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4 \). Then, calculate \( (0.25)^3 = 0.015625 \) and \( (0.75)^1 = 0.75 \). Multiply these results together: \( P(3) = 4 \times 0.015625 \times 0.75 = 0.046875 \).

Key Concepts

Probability Formula

In statistics, the binomial probability formula helps us find the likelihood of a particular outcome, like flipping a coin and getting heads a certain number of times. This formula is crucial when you're looking at situations where there are fixed probabilities for success or failure in repeated trials. It looks like this: \[ P(x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \]Here's a quick guide:

• \( n \) is the number of trials you perform. Think of it as the number of times you're trying something.
• \( x \) refers to the desired number of successful outcomes you want to achieve in your trials.
• \( \pi \) is the probability of achieving a success in any single trial.
• \( (1-\pi) \) is the probability of failure in any single trial.

This formula multiplies together three main parts: the number of ways the event can happen, the probability of each successful outcome, and the probability of each failed outcome.
Just remember, the formula gives you the chance of **exactly** \( x \) successes in \( n \) trials, not at least or at most \( x \). It's a specific, targeted calculation.

Binomial Coefficient

The binomial coefficient, represented by \( \binom{n}{x} \), is an important part of the binomial probability formula. It calculates how many ways you can achieve \( x \) successes out of \( n \) trials. In simple terms, it's all about combinations.The formula for the binomial coefficient is:\[\binom{n}{x} = \frac{n!}{x!(n-x)!}\]Here's what each symbol means:

• \( n! \) ("n factorial") is the product of all positive integers up to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
• \( x! \) is the factorial of \( x \) and follows the same pattern.
• \( (n-x)! \) is the factorial of the difference between \( n \) and \( x \).

The binomial coefficient answers the question: "In how many different ways can we select \( x \) items from \( n \) available options?" For calculations, it's simpler than it might seem - just substitute your numbers into the formula and compute. In our example cases:

• \( \binom{4}{2}=6 \) means there are 6 unique ways to choose 2 successes from 4 trials.
• \( \binom{4}{3}=4 \) gives us 4 ways to choose 3 successes from 4 trials.

This calculation is central to determining the actual probability of your outcomes.

Number of Trials

The "number of trials" is a foundational concept in any binomial probability scenario. It represents the total attempts you have in the experiment (or the number of times you perform a given action). In the given exercise, this is represented as \( n = 4 \).To visualize this, imagine you are rolling a die 4 times. In this situation:

• Each roll of the die is a single trial.
• The collection of all 4 rolls forms your total number of trials.

The "number of trials" element tells you how extensive your scenario is. Without a specified number of trials, we can't precisely calculate probabilities, because we wouldn't know how many times the event or action is being tested.It's key to remember that every trial is independent, which means the outcome of one doesn't directly affect another. In our example, regardless of what happens in one die roll, the probability remains the same for each subsequent roll.Knowing the number of trials allows us to plug \( n \) into the binomial probability formula and compute the likelihood of seeing a set number of successes over these trials. This concept is instrumental in understanding how outcomes distribute over a series of events.