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 that at least one insect survives is approximately 0.651.

Step 1: Understanding Complement Probability

First, recognize that the probability of an individual insect not surviving past five days is the complement of surviving: if an insect has a 0.1 probability of living more than five days, the probability of it not surviving past five days is \(1 - 0.1 = 0.9\).

Step 2: Calculate Probability of None Surviving

Calculate the probability that none of the 10 insects is alive after five days (i.e., all of them die). This is a binomial probability calculation where all 10 insects do not survive beyond five days: \(P(\text{none survive}) = (0.9)^{10}\).

Step 3: Complement to Find At Least One Survives

The probability of at least one insect surviving is the complement of the probability that none survive: \(P(\text{at least one survives}) = 1 - P(\text{none survive})\).

Step 4: Substitute and Solve

Substitute the probability found in Step 2 into the equation from Step 3: \(P(\text{at least one survives}) = 1 - (0.9)^{10}\). Calculate \((0.9)^{10}\) and then subtract from 1 to find the final result.

Key Concepts

Understanding Complement Probability

When dealing with probabilities, it's important to recognize the concept of a complement. The complement of an event is essentially the opposite outcome of that event. In this scenario, the probability of an insect surviving beyond five days is 0.1. Therefore, the complement, which is the probability of an insect *not* surviving beyond five days, is calculated by subtracting the probability from 1. Thus:

• The survival probability is 0.1
• The non-survival probability is then: \(1 - 0.1 = 0.9\)

Understanding complement probability helps determine the likelihood of various scenarios, such as none or at least one of the insects surviving beyond a certain period.

Efficient Probability Calculation

Calculating probabilities involves identifying the likelihood of a particular event occurring, based on known information. In our specific case of 10 insects, using the complement probability makes calculations much easier. Instead of calculating for each possible number of survivors, focusing on none surviving can simplify the work greatly. For the group, we assume each insect has a 0.9 probability of dying after five days. Therefore, the probability of none of the insects surviving (i.e., all dying) is calculated as:- \(P(\text{none survive}) = (0.9)^{10}\)Ultimately, the simplicity and efficiency of these calculations in large samples highlight the power of the complement rule in probability.

Importance of Sample Size in Probability

The concept of sample size is crucial in probability as it affects the accuracy and reliability of the results. A bigger sample size often provides more reliable results as it reduces the impact of random anomalies. Here, the sample size is 10 insects. This is significant because: - The larger the sample, the more likely the calculated probability reflects the true population probability - It reduces the role of chance errors In smaller samples, individual variations have a bigger impact on results, potentially skewing the probability calculations. Hence, understanding sample size helps interpret probability results more accurately.

Probability of Survival

Ultimately, the goal is to determine the probability that at least one insect survives. The previously calculated complement probability allows us to find this easily. If the probability that none survive is found using \((0.9)^{10}\), the probability of at least one surviving becomes:- \(P(\text{at least one survives}) = 1 - (0.9)^{10}\)This approach simplifies understanding survival probability in any context. Whether it's understanding the likelihood of an organism surviving or predicting how many might endure certain conditions, seeing survival through probabilities helps in planning and decision-making.