Problem 41 - Calculus for Biology and Medicine

Question

Assume that \(20 \%\) of all plants in a field are infested with aphids. Suppose that you pick 20 plants at random. What is the probability that none of them carried aphids?

Step-by-Step Solution

Answer

The probability that none of the 20 plants are infested is approximately 0.0115.

Step 1: Understand the problem

We are given that 20% of the plants in a field are infested with aphids. We need to find the probability that none of the 20 randomly picked plants are infested.

Step 2: Identify the probability of an individual plant not being infested

Since 20% of plants are infested, the probability that a plant is not infested is the complement of that, which is \(1 - 0.2 = 0.8\).

Step 3: Calculate the probability for 20 plants

The probability that all 20 plants are not infested is the product of each plant not being infested. Therefore, we use the formula for independent events: \((0.8)^{20}\).

Step 4: Compute the probability

Calculate \((0.8)^{20}\) using a calculator: \((0.8)^{20} \approx 0.0115\).

Key Concepts

Understanding Binomial Probability

Binomial probability is a fundamental concept in probability theory. It involves any process that has two possible outcomes. In the case of the plant infestation problem, each plant either is infested or it isn't.
The binomial probability formula helps us determine the likelihood of having a certain number of successes in a set of independent trials. A success in this context could mean a plant is *not* infested, given that finding an uninfested plant fits our desired outcome here.
Typically, the formula for binomial probability is expressed as: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where:

• \(n\) is the number of trials (plants picked)
• \(k\) is the number of desired successful outcomes (plants not infested)
• \(p\) is the probability of success on an individual trial
• \(\binom{n}{k}\) is a binomial coefficient

In our case, we're calculating the odds of zero infested plants, which simplifies the solution, as seen in the step-by-step solution.
Understanding these concepts will allow you to tackle similar problems with varying levels of complexity.

The Complement Rule in Probability

The complement rule is a simple yet powerful tool in probability theory. It hinges on the idea that the probability of an event occurring is 1 minus the probability that it does not occur.
In the context of the aphid problem, we know that 20% of plants are infested. Thus, the probability of picking a plant that is not infested is the complement: \[P( ext{not infested}) = 1 - P( ext{infested}) = 1 - 0.2 = 0.8 \]
This rule can simplify calculations significantly, especially when determining the probability of an event happening multiple times, such as selecting 20 plants none of which are infested.
Using the complement rule ensures accuracy and reduces the time spent on unnecessarily complex calculations. Understanding how complements work in probability is essential for tackling problems like our example and others with more intricate scenarios.

Independent Events in Probability

The concept of independent events in probability refers to a situation where the occurrence of one event does not affect the likelihood of another event occurring. This is a crucial assumption in many probability problems.
For the 20 plants in the field, assuming independence means that removing one plant does not change the probability of the next plant being infested. In mathematical terms, the probability of any specific plant being uninfested remains constant, independent of the others, calculated as \[P( ext{not infested}) = 0.8\].
For independent events, the probability of all events occurring is the product of their individual probabilities. Thus, the independent probability of all 20 plants being uninfested is \[(0.8)^{20}\].
This independence simplifies the calculation significantly and helps apply the binomial probability model effectively. Recognizing when events are independent is key to applying probability rules correctly and efficiently.