Question

A wildlife biologist examines frogs for a genetic trait he suspects may be linked to sensitivity to industrial toxins in the environment. Previous research had established that this trait is usually found in 1 of every 8 frogs. He collects and examines a dozen frogs. If the frequency of the trait has not changed, what's the probability he finds the trait in a) none of the 12 frogs? b) at least 2 frogs? c) 3 or 4 frogs? d) no more than 4 frogs?

Short answer

a) 0.257 b) 0.376 c) 0.273 d) 0.956

Step-by-step solution

Identify the Parameters

The problem involves a binomial distribution. We have a sample size of 12 frogs (n=12), and the probability of finding the genetic trait in a single frog is 1/8 (p=0.125). The complementary probability, not finding the trait, is q=0.875.

Calculate Probability for Part a

We need to find the probability of finding the trait in none of the 12 frogs (k=0). Using the binomial probability formula: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]For k=0, the formula simplifies to: \[ P(X=0) = \binom{12}{0} (0.125)^0 (0.875)^{12} = (0.875)^{12} \]Calculate \((0.875)^{12} \approx 0.257\).

Calculate Probability for Part b

We need the probability of finding the trait in at least 2 frogs, which is \( P(X \geq 2) \). This is the complement of finding 0 or 1 frog with the trait: \[ P(X \geq 2) = 1 - P(X=0) - P(X=1) \]We already found \( P(X=0) \). Now calculate \( P(X=1) \) using:\[ P(X=1) = \binom{12}{1} (0.125)^1 (0.875)^{11} \approx 0.367 \]Then, subtract from 1:\[ P(X \geq 2) = 1 - 0.257 - 0.367 = 0.376 \]

Calculate Probability for Part c

Find the probability of finding the trait in exactly 3 or 4 frogs.\[ P(X=3) = \binom{12}{3} (0.125)^3 (0.875)^9 \approx 0.197 \]\[ P(X=4) = \binom{12}{4} (0.125)^4 (0.875)^8 \approx 0.076 \]Add these probabilities together:\[ P(3 \text{ or } 4) = 0.197 + 0.076 = 0.273 \]

Calculate Probability for Part d

We need the probability of finding the trait in no more than 4 frogs, \( P(X \leq 4) \). This is the sum of the probabilities from 0 to 4 frogs.We already found \( P(X=0) \), \( P(X=1) \), \( P(X=3) \), and \( P(X=4) \). We still need \( P(X=2) \):\[ P(X=2) = \binom{12}{2} (0.125)^2 (0.875)^{10} \approx 0.283 \]Sum these probabilities:\[ P(X \leq 4) = 0.257 + 0.367 + 0.283 + 0.197 + 0.076 = 0.956 \]

Key concepts

Probability

Probability is the measure of the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). In our wildlife biologist scenario, we're interested in the probability of finding frogs with a specific genetic trait. When dealing with multiple possible outcomes, such as the number of frogs showing a genetic trait, probability can be understood through distributions. The binomial distribution is particularly helpful here because it considers the number of successes in a series of independent and identical experiments.When determining probabilities for the number of frogs with the trait, we use the binomial probability formula: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]In this formula, \( n \) is the number of trials (12 frogs), \( k \) is the number of successes (frogs with the trait), and \( p \) is the probability of success in a single trial (1/8 or 0.125). Understanding these parameters allows us to compute various probabilities, helping evaluate genetic traits in the population.

Genetic Trait Analysis

Genetic trait analysis is the study of genetic characteristics within a population to understand how often certain traits appear and their possible implications. In our frog scenario, a specific genetic trait may affect the frogs' sensitivity to industrial toxins. This type of analysis is crucial for:

• Determining the prevalence of genetic traits that might impact survival
• Assessing the adaptive abilities of organisms in changing environments
• Identifying traits linked to environmental stressors, like toxins

The steps involved often include collecting data from samples (like the dozen frogs), identifying traits of interest, and applying statistical methods, such as binomial distribution, to infer broader trends.

Industrial Toxins

Industrial toxins are pollutants released from industrial activities, potentially harming living organisms and ecosystems. These chemicals can affect wildlife in numerous ways, leading to health issues such as genetic mutations or heightened sensitivity to environmental changes. In the context of our problem:

• The identified genetic trait might make frogs more susceptible to industrial toxins.
• Understanding the distribution of this trait helps gauge how many frogs could be at risk.

Monitoring genetic traits linked to toxin sensitivity is essential for wildlife conservation and developing strategies to reduce pollutants' impact. This process underscores the importance of connecting genetic data with environmental health.

Wildlife Biology

Wildlife biology focuses on the study of animals, their behaviors, and interactions with ecosystems, including how they adapt to environmental changes. This field encompasses genetics, ecology, and conservation, equipping us to maintain biodiversity and ecological health. In practice, wildlife biologists might:

• Conduct field studies, like examining frogs for traits
• Analyze how traits affect survival or reproduction
• Use statistics to predict and manage wildlife populations

Understanding a genetic trait's prevalence gives insight into how wildlife might cope with external pressures, such as industrial toxins. Conservation strategies often rely on these biological assessments to ensure sustainable environments for diverse species.