Problem 25 - Algebra and Trigonometry

Question

Sick Leave The probability that a given worker at the Dyno Nutrition will call in sick on a Monday is \(0.04 .\) The packaging department has 8 workers. What is the probability that 2 or more packaging workers will call in sick next Monday?

Step-by-Step Solution

Answer

0.0186

Step 1: Understand the scenario

We are dealing with a probability problem involving binomial distribution because each worker has a fixed probability of calling in sick, independently of others, and there are a fixed number of trials (8 workers).

Step 2: Identify the parameters

We have a probability of calling in sick for a single worker as \(p = 0.04\), number of workers \(n = 8\), and we want to find the probability for 2 or more workers calling in sick.

Step 3: Approach using the complement rule

Instead of directly calculating the probability of 2 or more workers calling in sick (\(P(X \geq 2)\)), it is often easier to calculate \(P(X < 2)\) and then use the complement rule: \(P(X \geq 2) = 1 - P(X < 2)\).

Step 4: Calculate \(P(X = 0)\)

Using the binomial probability formula, \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\). For \(X = 0\), the formula becomes \(P(X = 0) = \binom{8}{0} (0.04)^0 (0.96)^8 = 0.7364\).

Step 5: Calculate \(P(X = 1)\)

Apply the binomial probability formula for \(X = 1\): \(P(X = 1) = \binom{8}{1} (0.04)^1 (0.96)^7 = 0.2450\).

Step 6: Find \(P(X < 2)\)

Combine the probabilities: \(P(X < 2) = P(X = 0) + P(X = 1) = 0.7364 + 0.2450 = 0.9814\).

Step 7: Use complement rule

Using the result from \(P(X < 2)\), calculate \(P(X \geq 2)\): \(P(X \geq 2) = 1 - 0.9814 = 0.0186\).

Key Concepts

Complement Rule

When dealing with probability, especially in scenarios involving more complex calculations, the complement rule can be incredibly useful. This rule provides an alternative way to calculate probabilities by focusing on the opposite outcome of what you're interested in finding. Instead of calculating the probability of an event directly, you calculate the probability of the complementary event, which is usually simpler, and then subtract it from 1. In our case, finding the probability of 2 or more workers calling in sick (\(P(X \geq 2)\)) can be complicated, but finding the probability of fewer than 2 workers calling in sick (\(P(X < 2)\)) is more straightforward. Once calculated, simply subtract this result from 1 to find your answer. Think of the complement rule as asking, "What is the opposite of the event we're interested in?"
Use it whenever direct computations seem cumbersome, and you'll often save time and effort:

• Identify the event of interest.
• Define the complementary event.
• Calculate the probability of the complementary event.
• Subtract this probability from 1 to find the probability of the original event.

Probability Calculation

Calculating probabilities, especially in binomial distributions, can seem tricky at first, but breaking it down step by step makes it manageable. In binomial probability scenarios, you're typically dealing with a fixed number of trials and a fixed probability for each trial. In our exercise, "trials" represent the workers, each with a chance of calling in sick or not.
To perform a probability calculation:

• Identify the total number of trials (\(n\)) and the probability of success (\(p\)). Success, in this context, is a worker calling in sick.
• Determine what probability you need to calculate. Are you calculating zero occurrences, one, or two or more?
• Utilize the binomial probability formula to compute the specific probabilities for results of zero or one, based on the complement rule.
• Add the calculated probabilities for combined events, like calculating \(P(X < 2)\) by adding \(P(X = 0)\) and \(P(X = 1)\).

Remember, proficiency in probability calculation can be beneficial in various fields, so practicing these skills is well worth the effort.

Binomial Probability Formula

The binomial probability formula is a key tool in probability theory, especially for scenarios like the one we're examining. This formula allows you to calculate the likelihood of obtaining a specific number of "successes" in a set number of independent trials, where the outcome of each trial is binary (like success/failure or sick/not sick). The formula is as follows: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

• \(\binom{n}{k}\): This is the binomial coefficient, representing the number of ways to choose \(k\) successes from \(n\) trials.
• \(p\^k\): Here, \(p\) is the probability of success on a single trial, raised to the power of \(k\), the number of successes.
• \((1-p)\^{n-k}\): This expresses the probability of failure (1-p), raised to the power of the remaining trials, \(n-k\).

For example, to find \(P(X = 0)\) in our scenario, calculate each part and multiply them together, using \(n = 8\), \(k = 0\), and \(p = 0.04\). By mastering this formula, you can tackle a wide range of problems involving binomial probabilities.