Problem 20 - Basic Statistics for Business & Economics

Question

In a binomial distribution, \(n=12\) and \(\pi=.60 .\) Find the following probabilities. a. \(x=5\). b. \(x \leq 5\) c. \(x \geq 6\)

Step-by-Step Solution

Answer

(a) 0.1010, (b) 0.1185, (c) 0.8815

Step 1: Understand the Binomial Distribution

A binomial distribution is defined by two parameters: \(n\), the number of trials, and \(\pi\), the probability of success in each trial. In this case, \(n=12\) and \(\pi=0.60\). The formula for the probability of exactly \(x\) successes is given by \(P(X=x) = \binom{n}{x} \pi^x (1-\pi)^{n-x}\).

Step 2: Compute P(X=5)

To find \(P(X=5)\), substitute \(n=12\), \(\pi=0.60\), and \(x=5\) into the binomial formula: \[P(X=5) = \binom{12}{5} (0.60)^5 (0.40)^7\]\[= \frac{12!}{5!(12-5)!} \times 0.07776 \times 0.0016384\]\[= 792 \times 0.000127\]\[= 0.1010\]

Step 3: Compute P(X≤5)

To find \(P(X \leq 5)\), you calculate the sum of probabilities from \(x=0\) to \(x=5\): \[P(X \leq 5) = \sum_{x=0}^{5} \binom{12}{x} (0.60)^x (0.40)^{12-x}\] You compute each of these terms using the binomial formula and add them up:\[P(X=0) = 0.000002\]\[P(X=1) = 0.000032\]\[P(X=2) = 0.000442\]\[P(X=3) = 0.003157\]\[P(X=4) = 0.013799\]\[P(X=5) = 0.101032\]\[P(X \leq 5) = 0.000002 + 0.000032 + 0.000442 + 0.003157 + 0.013799 + 0.101032 = 0.118464\]

Step 4: Compute P(X≥6)

Since \(P(X \geq 6) = 1 - P(X \leq 5)\), where \(P(X \leq 5)\) was computed in the previous step:\[P(X \geq 6) = 1 - 0.118464\]\[= 0.881536\]

Key Concepts

Probability of Success

The probability of success is a fundamental parameter in binomial distributions and plays a pivotal role in determining the behavior of the distribution. In any given trial within a binomial setting, an event is labeled as a success when it meets the predetermined criteria for success. The probability of this occurrence is represented by \( \pi \).
For example, consider a situation where we want to know the probability of a basketball player making a free throw, and historically, the player hits the basket 60% of the time. This 60% or 0.60 is our probability of success, \( \pi \). In the problem we're considering, \( \pi = 0.60 \) which implies that each individual trial has a 60% chance of being successful.

It's crucial to understand that the probability of success in a binomial experiment remains constant across all trials. This is one of the key distinguishing characteristics of binomial distributions, ensuring a uniform probability model throughout.

Binomial Probability Formula

The binomial probability formula is the mathematical tool we use to determine the likelihood of achieving exactly \( x \) successes in \( n \) trials of a binomial experiment. The formula is presented as:

\[ P(X=x) = \binom{n}{x} \pi^x (1-\pi)^{n-x} \]Here, \( \binom{n}{x} \) represents the binomial coefficient, which calculates the number of ways to choose \( x \) successes out of \( n \) trials. The \( \pi^x \) part of the formula calculates the probability of achieving \( x \) successes, while \( (1-\pi)^{n-x} \) calculates the probability of \( n-x \) failures.

Let's walk through an example to illustrate this. Using the binomial probability formula, if we want to find the probability of exactly 5 successes (\( x = 5 \)) out of 12 trials (\( n = 12 \)), where each trial has a probability of success \( \pi = 0.60 \), we substitute these values into the formula. Calculating each component step by step gives us:
1. **Calculate the binomial coefficient**: \( \binom{12}{5} = \frac{12!}{5!(12-5)!} = 792 \)2. **Probability of 5 successes**: \( (0.60)^5 = 0.07776 \)3. **Probability of 7 failures**: \( (0.40)^7 = 0.0016384 \)4. **Combine**: \( 792 \times 0.07776 \times 0.0016384 = 0.1010 \)

Thus, the probability of exactly 5 successes is 0.1010.

Cumulative Probability Calculation

Cumulative probability in a binomial distribution refers to calculating the probability of a certain number of successes or fewer, or the probability of that number or more. It is essentially summing up individual probabilities to find a collective chance.

To compute \( P(X \leq 5) \), for example, you need to calculate the sum of probabilities from \( x=0 \) to \( x=5 \) using the binomial probability formula for each individual \( x \). This involves calculating each probability step-by-step:- \( P(X=0) = 0.000002 \)- \( P(X=1) = 0.000032 \)- \( P(X=2) = 0.000442 \)- \( P(X=3) = 0.003157 \)- \( P(X=4) = 0.013799 \)- \( P(X=5) = 0.101032 \)Adding these up provides the cumulative probability \( P(X \leq 5) = 0.118464 \).

To find \( P(X \geq 6) \), instead of computing each probability from 6 to 12, you use the complementary rule:
- Calculate using: \( P(X \geq 6) = 1 - P(X \leq 5) \).
- Since \( P(X \leq 5) = 0.118464 \), it follows that \( P(X \geq 6) = 1 - 0.118464 = 0.881536 \).

Cumulative probability calculations are efficient and make predicting ranged outcomes feasible in binomial distributions.