Problem 10 - Basic Statistics for Business & Economics

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: Understand the Binomial Formula

The binomial probability formula is used to find the probability of achieving exactly \( x \) successes in \( n \) trials with a success probability of \( \pi \) on each trial. The formula is given by: \[ P(X = x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \] where \( \binom{n}{x} \) is the binomial coefficient calculated as \( \frac{n!}{x!(n-x)!} \).

Step 2: Calculate for x=1

Substitute \( x = 1 \), \( n = 5 \), and \( \pi = 0.4 \) into the binomial formula: \[ P(X = 1) = \binom{5}{1} (0.4)^1 (0.6)^4 \]Calculate the binomial coefficient \( \binom{5}{1} = 5 \). Thus: \[ P(X = 1) = 5 \times 0.4 \times 0.6^4 \]Calculate \( 0.6^4 = 0.1296 \) and substitute it back: \[ P(X = 1) = 5 \times 0.4 \times 0.1296 = 0.2592 \]

Step 3: Calculate for x=2

Similarly, substitute \( x = 2 \), \( n = 5 \), and \( \pi = 0.4 \) into the binomial formula: \[ P(X = 2) = \binom{5}{2} (0.4)^2 (0.6)^3 \]Calculate the binomial coefficient \( \binom{5}{2} = 10 \). Thus: \[ P(X = 2) = 10 \times 0.16 \times 0.216 \] Compute the values: \( 0.4^2 = 0.16 \) and \( 0.6^3 = 0.216 \). Substitute these back to get: \[ P(X = 2) = 10 \times 0.16 \times 0.216 = 0.3456 \]

Key Concepts

Probability Calculation

Probability calculations allow us to measure the likelihood of an event happening. In the context of binomial distribution, we're often interested in the probability of a certain number of successes in repeated trials. This involves understanding how likely it is for a particular outcome to occur.

To do this in a binomial experiment, you start by identifying the probability of success (\( \pi \)) and consider the total number of trials (\( n \)). The probability of achieving exactly \( x \) successes is calculated using the binomial formula.

Remember, a probability value lies between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. This calculation helps us quantify specific outcomes, like determining the probability of getting exactly 2 heads when tossing a coin 5 times. It's a valuable tool in decision-making processes by providing a mathematical basis for predictions.

Binomial Formula

The binomial formula is a powerful tool used in statistics to find the probability of \( x \) successes in \( n \) trials. This formula is especially handy when dealing with binary outcomes—outcomes that can be classified into two categories such as success or failure. For the given probability scenario, the formula is written as: \[P(X = x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \] where:

• \( \binom{n}{x} \) is the binomial coefficient
• \( \pi \) is the probability of success on a single trial
• \( (1-\pi) \) is the probability of failure

By substituting the known values into the formula, you can calculate the probability of achieving a predefined number of successes in a series of trials. It's important to plug in accurate values of \( n \) and \( \pi \) to obtain precise results. The beauty of the binomial formula lies in its ability to adapt to various probability scenarios, making it a cornerstone in probability theory.

Binomial Coefficient

The binomial coefficient is a fundamental part of calculating probabilities in a binomial distribution. Often referred to as "combinations," this coefficient tells us the number of ways to choose \( x \) successes from \( n \) trials. The mathematical notation for binomial coefficient is\( \binom{n}{x} \), and it is computed using the formula:
\[\binom{n}{x} = \frac{n!}{x!(n-x)!}\]
where \( n! \) represents factorial of \( n \), meaning you multiply \( n \) by every whole number below it down to 1.

For instance, when calculating \( \binom{5}{2} \), you would compute as follows:
\[\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]
This means there are 10 possible ways to choose 2 successes out of 5 trials. The binomial coefficient is essential in determining the respective probabilities of different outcomes in a binomial experiment.