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

(a) 1 spoiled apple per box; (b) about 3.5 boxes with no spoiled apples.

Step 1: Understand the Problem

We have a box with 10 apples, and each apple has a probability of 0.1 of being spoiled. We need to find the expected number of spoiled apples in one box and then the expected number of boxes with no spoiled apples in a shipment of 10 boxes.

Step 2: Calculate Expected Number of Spoiled Apples Per Box

The expected number of spoiled apples in a box is calculated by multiplying the number of apples by the probability of spoilage. Since the probability of spoilage is 0.1 and there are 10 apples, the expected number is: \( E(X) = 10 \times 0.1 = 1 \).Thus, on average, we expect 1 spoiled apple per box.

Step 3: Compute Probability of No Spoiled Apples in One Box

To find the probability that a box contains no spoiled apples, we use the binomial probability formula. The probability of no apples being spoiled (\( k = 0 \)) is:\[ P(X=0) = \binom{10}{0} (0.1)^0 (0.9)^{10} = (0.9)^{10} \].Calculating \((0.9)^{10}\) gives approximately 0.3487.

Step 4: Calculate Expected Number of Boxes with No Spoiled Apples

In a shipment of 10 boxes, the expected number of boxes with no spoiled apples is the total number of boxes multiplied by the probability of no spoiled apples in one box:\[ E(Y) = 10 \times 0.3487 = 3.487 \].This means, on average, about 3.487 boxes are expected to have no spoiled apples in a shipment of 10.

Key Concepts

Expected Value

The expected value is a concept rooted in probability and statistics that represents the average or mean value you would expect in a certain scenario if you repeated it multiple times. It gives you a way to predict outcomes without having to conduct actual experiments each time.

In our apple box scenario, where each apple has a probability of being spoiled as 0.1, we calculate the expected number of spoiled apples per box by multiplying the probability of spoilage by the total number of apples. Using the formula:

• Expected Value = Number of Trials × Probability of Success
• \[ E(X) = 10 \times 0.1 = 1 \]

This tells us that, on average, we can expect 1 spoiled apple per box, which provides a meaningful way to anticipate what is likely to happen on average when dealing with multiple boxes.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states across a number of trials. It’s particularly useful when you're dealing with questions regarding the number of successes in a series of independent experiments.

For instance, in our example with the apples, we are determining the probability of a particular number of apples being spoiled in a box of 10 apples. This scenario fits well with the binomial distribution because:

• Each apple can either be spoiled or not (success or failure).
• The probability of spoilage is constant at 0.1 for each apple.
• The spoilage of one apple does not affect the other (independent events).

We compute the probability of having no spoiled apples in one box using:

• \[ P(X=0) = \binom{10}{0} (0.1)^0 (0.9)^{10} = (0.9)^{10} \approx 0.3487 \]

This distribution helps us understand variations and the likelihood of different outcomes when similar conditions are maintained.

Independent Events

In probability, independent events are events where the occurrence of one event does not affect the occurrence of another. This property simplifies calculations and assumptions in probability theory.

In the context of the apple spoilage problem, the spoilage of each apple is an independent event. This means:

• The spoilage of one apple does not influence whether another apple is spoiled.
• Each spoilage event has the same probability of \,0.1\,, regardless of what happens to the other apples.

Understanding independence is critical when using distributions like the binomial because it justifies our ability to multiply probabilities across trials.

In the shipment of 10 boxes problem, independence allows us to compute the expected number of boxes with no spoiled apples using:

• \[ E(Y) = 10 \times (0.9)^{10} \approx 3.487 \]

Here, independence ensures the probability calculation remains valid across all boxes in the shipment.