Problem 35 - Calculus for Biology and Medicine

Question

Assume that the probability that an insect species lives more than five days is \(0.1\). Find the probability that, in a sample of size 10 of this species, at least one insect will still be alive after five days.

Step-by-Step Solution

Answer

The probability is approximately 0.6513.

Step 1: Identify Parameters

We are given that each insect has a 0.1 probability of surviving more than five days. Additionally, we are working with a sample of size 10. We can define this scenario as a binomial probability problem, where the number of trials is 10 and the probability of success (an insect surviving more than five days) is 0.1.

Step 2: Define the Event

We are interested in finding the probability that at least one insect out of 10 survives more than five days. This is the complement of the probability that none survive more than five days.

Step 3: Calculate Complementary Probability

Calculate the probability that none of the 10 insects survive more than 5 days. This is a binomial probability expression where the number of successes (insects surviving) is zero:\[ P(X=0) = \binom{10}{0} \cdot (0.1)^0 \cdot (0.9)^{10} = (0.9)^{10} \]

Step 4: Find the Desired Probability

The probability that at least one insect survives more than five days is the complement of the probability that none survive:\[ P(X \geq 1) = 1 - P(X=0) = 1 - (0.9)^{10} \]

Step 5: Calculate Final Probability

Calculate the value:\[ (0.9)^{10} \approx 0.3487 \]Thus, \[ P(X \geq 1) = 1 - 0.3487 = 0.6513 \]

Key Concepts

Probability Theory Basics

Probability theory is a branch of mathematics that studies the likelihood of events occurring. When working with probabilities, these values are expressed between 0 and 1:

• A probability of 0 means an event cannot happen.
• A probability of 1 means an event is certain to happen.

In probability theory, our goal is often to determine how likely it is for a particular event to occur given certain conditions. For example, if we know the chance of a specific insect living beyond five days, we can use this information to calculate other probabilities related to a group of these insects. Essentially, probability helps us quantify uncertainty and make informed predictions.
In our exercise, we are interested in knowing the likelihood of at least one insect surviving more than five days, given the probability and sample size. This is a classic example of applying probability theory to practical problems.

Understanding Complementary Probability

Complementary probability is a concept used to simplify the calculation of probabilities. It is often much easier to calculate the probability of the complement event (the opposite of what we are interested in) and then subtract it from 1.
Here's the idea: If you're trying to find the probability of event A happening and it's hard to compute directly, you can find the probability of event A not happening (complement of A), and then subtract that from 1. This is because in a complete set of possibilities, the event happening and not happening covers all outcomes:

• Probability of A happening = 1 - Probability of A not happening.

For instance, in our scenario, we wanted to find out the probability that at least one insect survives. It was easier to calculate the probability that no insects survive, which is a direct application of complementary probability, and subtract this from 1 to find our desired solution.

Understanding Binomial Probability Expressions

The binomial probability expression comes into play when dealing with scenarios where there are a fixed number of independent trials, each with two possible outcomes (success or failure). One common way to express this is by using combinations and probabilities.
In a binomial probability setting:

• Each trial has the same probability of success.
• The number of successful outcomes in a fixed number of trials can be described using a binomial coefficient.

For example, when looking for the probability that exactly k successes occur in n trials, the binomial probability formula is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Where:

• \(n\) is the number of trials.
• \(k\) is the number of successes (which could be zero for complete failure scenarios).
• \(p\) is the probability of success on each trial.

In the example, our interest was in scenarios involving no insects surviving past five days (zero successes). Using the expression \( (0.9)^{10} \), we found this specific probability and then applied complementary probability to find the probability of at least one success (survival).
This demonstrates how the concepts of binomial probability expressions and complementary probability nicely come together to solve real-world problems.