Problem 19 - Basic Statistics for Business and Economics with Student

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: 0.2668, b: 0.5214, c: 0.4786

Step 1: Understand the Binomial Distribution

The binomial distribution is used for a fixed number of independent trials, each having two possible outcomes: 'success' and 'failure'. Here, we have 8 trials (\(n = 8\)) and the probability of success (\(\pi\)) is 0.30.

Step 2: Use the Binomial Probability Formula

The probability of exactly \(x\) successes in \(n\) trials is given by: \[P(X=x) = \binom{n}{x} \cdot \pi^x \cdot (1-\pi)^{n-x}\]We will apply this formula for \(x=2\), \(x \leq 2\), and \(x \geq 3\).

Step 3: Calculate Probability for x=2

Using the formula, calculate:\[P(X=2) = \binom{8}{2} \cdot (0.30)^2 \cdot (0.70)^6\]First, find \(\binom{8}{2} = \frac{8 \cdot 7}{2 \cdot 1} = 28\). Then calculate:\[(0.30)^2 = 0.09\] and\[(0.70)^6 \approx 0.118\]So, \[P(X=2) \approx 28 \times 0.09 \times 0.118 \approx 0.2668\]

Step 4: Calculate Probability for x <= 2

Find the sum of probabilities for \(x=0\), \(x=1\), and \(x=2\):For \(x=0\):\[P(X=0) = \binom{8}{0} \cdot (0.30)^0 \cdot (0.70)^8 = 1 \cdot 1 \cdot 0.0576 = 0.0576\]For \(x=1\):\[P(X=1) = \binom{8}{1} \cdot (0.30)^1 \cdot (0.70)^7 = 8 \cdot 0.30 \cdot 0.0824 = 0.197\]Sum these probabilities:\[P(X\leq2) = 0.0576 + 0.197 + 0.2668 = 0.5214\]

Step 5: Calculate Probability for x >= 3 Using Complement Rule

Using the complement rule, \[P(X \geq 3) = 1 - P(X \leq 2)\]From Step 4, we know \[P(X \leq 2) = 0.5214\]Thus, \[P(X \geq 3) = 1 - 0.5214 = 0.4786\]

Key Concepts

Binomial Probability Formula

The binomial probability formula is a powerful tool in statistics that helps us find the probability of exactly \(x\) successes in \(n\) independent trials. Each trial has two possible outcomes: success with probability \(\pi\) and failure with probability \(1-\pi\). The formula is:\[P(X=x) = \binom{n}{x} \cdot \pi^x \cdot (1-\pi)^{n-x}\]Where:

• \(\binom{n}{x}\) is the binomial coefficient, calculated as \(\frac{n!}{x!(n-x)!}\), which counts how many ways we can choose \(x\) successes out of \(n\) trials.
• \(\pi^x\) gives the probability of getting \(x\) successes.
• \((1-\pi)^{n-x}\) accounts for the probability of the remaining \(n-x\) being failures.

In our exercise, we have 8 trials with a 0.30 probability of success. To find specific probabilities like \(P(X=2)\), plug \(n=8\), \(x=2\), and \(\pi=0.30\) into the formula to get the probability.

Probability Calculations

To calculate the probability for \(x\) successes, follow the step of inserting values into the binomial probability formula. For example, if you want to find \(P(X=2)\), substitute \(n=8\), \(x=2\), and \(\pi=0.30\). Calculating step-by-step:1. Calculate the binomial coefficient \(\binom{8}{2} = 28\).2. Compute \((0.30)^2 = 0.09\).3. Calculate \((0.70)^6 \approx 0.118\).4. Multiply these values together: \(28 \times 0.09 \times 0.118 \approx 0.2668\).This gives us \(P(X=2) \approx 0.2668\). Similarly, to find the cumulative probability \(P(X \leq 2)\), sum up probabilities for \(x=0\), \(x=1\), and \(x=2\).
Each part involves plugging values into the formula and performing straightforward calculations. It's a systematic approach that makes calculating probabilities for discrete events manageable.

Complement Rule

The complement rule is a handy shortcut in probability calculations, especially when dealing with cumulative probabilities. The rule states that the probability of an event occurring is one minus the probability of the event not occurring. Mathematically: \[P(X \geq k) = 1 - P(X < k)\]In our case for \(P(X \geq 3)\), instead of calculating probabilities for several values directly, utilize:\[P(X \geq 3) = 1 - P(X \leq 2)\]From our previous calculation:

• We found \(P(X \leq 2) = 0.5214\).
• Therefore, \(P(X \geq 3) = 1 - 0.5214 = 0.4786\).

This method saves time and reduces potential errors by subtracting from one the cumulative probability of the complementary event.