Problem 43 - Calculus for Biology and Medicine

Question

Suppose that a box contains 10 apples. The probability that any one apple is spoiled is \(0.1\). (Assume that spoilage of the apples is an independent phenomenon.) (a) Find the expected number of spoiled apples per box. (b) A shipment contains 10 boxes of apples. Find the expected number of boxes that contain no spoiled apples.

Step-by-Step Solution

Answer

Expected spoiled apples per box is 1; expected boxes with no spoiled apples is about 3.487.

Step 1: Define the Random Variable

Let's denote the random variable \( X \) as the number of spoiled apples in one box. We know that each apple can either be good or spoiled, so it follows a Binomial distribution. The problem involves 10 trials (apples), each with a probability of spoilage \( p = 0.1 \). Thus, \( X \sim \text{Binomial}(n=10, p=0.1) \).

Step 2: Expected Value of Spoiled Apples in One Box

The expected value (or mean) for a Binomial random variable \( X \) is given by the formula \( E(X) = n \cdot p \). For this problem, \( n = 10 \) and \( p = 0.1 \), so \( E(X) = 10 \times 0.1 = 1 \). Thus, the expected number of spoiled apples per box is 1.

Step 3: Define the Problem for Boxes without Spoiled Apples

Now, let us define the random variable \( Y \) as the number of boxes without any spoiled apples. Here, the probability of having a box with no spoiled apples is equal to the probability of zero spoiled apples in one box.

Step 4: Calculate Probability of Zero Spoiled Apples in One Box

For a Binomial distribution, the probability of exactly \( k \) successes (spoiled apples) is given by \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \). For \( k=0 \) spoiled apples, \( P(X=0) = \binom{10}{0} (0.1)^0 (0.9)^{10} = (0.9)^{10} \). Calculate this value: \( (0.9)^{10} \approx 0.3487 \).

Step 5: Expected Number of Boxes with No Spoiled Apples

Now consider \( 10 \) boxes as the sum of 10 independent Bernoulli trials where each trial is the event that a box has zero spoiled apples (with probability \( 0.3487 \)). The expected number of success in Bernoulli trials is simply \( n \times \text{probability of success} \). Therefore, the expected number of boxes with no spoiled apples is \( 10 \times 0.3487 \approx 3.487 \).

Key Concepts

Binomial distribution

The Binomial distribution is a probability distribution that describes the outcome of a fixed number of independent trials. In each trial, there are two possible outcomes: success or failure. It's useful when you want to determine probabilities over multiple trials where the conditions do not change. The common real-world example is flipping a coin a set number of times. In the context of our problem, each apple is considered a 'trial,' with outcomes being either 'spoiled' (success) or 'not spoiled' (failure). You need two parameters to define a binomial distribution:

• Parameter \( n \) - the number of trials, which in our example is the number of apples per box, 10.
• Parameter \( p \) - the probability of one success (an apple being spoiled), which is given as 0.1.

The random variable \( X \) representing the number of spoiled apples follows a binomial distribution: \( X \sim \text{Binomial}(n=10, p=0.1) \).

Random variable

A random variable is a mathematical concept to describe outcomes in a probabilistic framework. It associates numerical values to each outcome of a random process. In our exercise, the random variable is \( X \), indicating how many apples are spoiled in one box of 10 apples.
The key characteristic of a random variable in this setting is that it can be both countable and discrete. Here, \( X \) can assume values from 0 to 10. Each of these values comes with a corresponding probability determined by the binomial distribution.
This concept helps us model and compute probabilities for different possible observations — like whether a box contains 2, 5, or no spoiled apples at all. Understanding the properties of random variables enables us to predict the outcomes based on probability theory.

Expected value

Expected value is a fundamental concept in probability, giving us the average outcome of a random variable over a large number of trials. It's computed using the probabilities of all possible outcomes multiplied by their respective values.
For a binomial random variable like \( X \), with parameters \( n \) and \( p \), the expected value, \( E(X) \), can be calculated simply by multiplying \( n \) and \( p \): \[E(X) = n \cdot p\]In the apple scenario, this translates to: \[E(X) = 10 \times 0.1 = 1\]This means that, on average, there is one spoiled apple per box. Expected value provides useful insights, especially when predicting outcomes over multiple trials or decisions, as it represents the "center" of a distribution.

Bernoulli trials

Bernoulli trials are the building blocks that underpin the binomial distribution. Each trial refers to a single experiment that can result in one of two outcomes: success or failure.

• A classic example is flipping a coin: heads represents success and tails failure.
• In our exercise, determining if one apple is spoiled represents a single Bernoulli trial, where 'spoiled' is considered a success.

In many scenarios, individual Bernoulli trials are repeated independently a certain number of times. Here, a box of apples represents independent trials since the spoilage of one apple does not affect the others.
Understanding Bernoulli trials helps set the foundation for working with binomial distributions and calculating probabilities. For example, when calculating the likelihood of multiple boxes having no spoiled apples, each box itself becomes a Bernoulli trial in a larger framework.