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Chapter 16: Problem 26

Frogs 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

Expert verified
a) Calculate \((\frac{7}{8})^{12}\). b) Subtract probabilities \(P(X=0)\) and \(P(X=1)\) from 1. c) Sum \(P(X=3)\) and \(P(X=4)\). d) Sum \(P(X=0)\) to \(P(X=4)\).

Step by step solution

01

Understanding the Probability

This problem involves calculating probabilities using the binomial distribution. The probability of finding a frog with the trait is given as \(p = \frac{1}{8}\). Since 1 out of 8 frogs have the trait, the probability that a frog does not have the trait is \(q = 1 - p = \frac{7}{8}\). We examine 12 frogs, thus \(n = 12\).
02

Probability of None of the 12 Frogs Having the Trait

a) To find the probability none of the 12 frogs have the trait, use the binomial probability formula: \(P(X = 0) = \binom{n}{0} p^0 q^{n-0}\), where \(n\) is 12, \(p\) is \(\frac{1}{8}\), and \(q\) is \(\frac{7}{8}\). Calculate: \(P(X=0) = \binom{12}{0} \left(\frac{1}{8}\right)^0 \left(\frac{7}{8}\right)^{12} = \left(\frac{7}{8}\right)^{12}\).
03

Probability of At Least 2 Frogs Having the Trait

b) The probability of at least 2 frogs having the trait is computed as \(P(X \geq 2) = 1 - P(X = 0) - P(X = 1)\). Calculate \(P(X = 1) = \binom{12}{1} \left(\frac{1}{8}\right)^1 \left(\frac{7}{8}\right)^{11}\). Then find: \(P(X \geq 2) = 1 - P(X=0) - P(X=1)\).
04

Probability of Exactly 3 or 4 Frogs Having the Trait

c) Calculate probabilities separately for \(X = 3\) and \(X = 4\):\[P(X=3) = \binom{12}{3} \left(\frac{1}{8}\right)^3 \left(\frac{7}{8}\right)^9\] and \[P(X=4) = \binom{12}{4} \left(\frac{1}{8}\right)^4 \left(\frac{7}{8}\right)^8\] Finally, add them: \(P(X = 3 \text{ or } 4) = P(X=3) + P(X=4)\).
05

Probability of No More Than 4 Frogs Having the Trait

d) "No more than 4 frogs" means we find the probability of 0, 1, 2, 3, or 4 frogs having the trait. Calculate each: \(P(X = 2) = \binom{12}{2} \left(\frac{1}{8}\right)^2 \left(\frac{7}{8}\right)^{10}\). Then sum: \(P(X \leq 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Probability calculation is a fundamental aspect of understanding how likely an event is to occur. In our frog example, we are interested in the probability that a certain genetic trait appears in a group of frogs. To compute this, we use the concept of binomial probability, which is suitable when each of several trials is independent of the others and has only two possible outcomes, like having or not having a trait.

Key components of probability calculation include:
  • Probability of success, denoted as \( p \), which in this case is the probability a frog has the genetic trait.
  • Probability of failure, \( q \), which is \( 1 - p \).
  • Number of trials, \( n \), representing the number of frogs examined.
Each calculation detail above helps to mathematically determine the likelihood of specific outcomes, using the rules laid out by the binomial distribution model.
Statistical Modeling
Statistical modeling involves applying mathematical systems to understand and predict outcomes within datasets. For our exercise, we apply statistical modeling to analyze genetic traits in frogs using assumptions based on prior research.

The core objective of statistical modeling here is to use the collected data of a dozen frogs and predict certain probabilities based on the binomial distribution model, helping to create an expected picture of probabilities in broader scenarios.
  • Incorporates prior research to inform probability values.
  • Predicts the number of frogs likely to exhibit a genetic trait.
  • Enables scientific conclusions based on probability calculations.
By meticulously modeling these parameters, biologists can better understand the presence of genetic traits and their implications.
Binomial Probability Formula
The binomial probability formula is crucial for calculating specific outcomes in experiments where there are two possible results, such as having a genetic trait or not.

The formula is expressed as follows:\[P(X = k) = \binom{n}{k} p^k q^{n-k}\]where:
  • \( n \) - number of trials (frogs examined)
  • \( k \) - number of successful outcomes (frogs with trait)
  • \( p \) - probability of success on a single trial
  • \( q \) - probability of failure on a single trial (\( q = 1-p \))
This formula helps determine probabilities for different scenarios, such as zero frogs having the trait or at least two, by summing or subtracting individual probabilities for specific outcomes.
Genetic Trait Analysis
Genetic trait analysis examines the presence and distribution of genetic characteristics within a population. It's essential for understanding implications for genetic diversity, adaptability, and sensitivity to environmental factors, like industrial toxins, in the case of our frog study.

Through probability calculations within a binomial distribution framework, scientists can assess how widespread a trait is and predict its presence in new samples without examining every individual.
  • Reveals potential links between genetic makeup and environmental influences.
  • Assists in identifying organisms at risk of environmental stressors.
  • Supports conservation efforts by highlighting ecological impacts.
Analyzing genetic traits through statistical methods thus supports informed conservation strategies and biological understanding.

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Most popular questions from this chapter

. Cold calls Justine works for an organization committed to raising money for Alzheimer's research. From past experience, the organization knows that about \(20 \%\) of all potential donors will agree to give something if contacted by phone. They also know that of all people donating, about \(5 \%\) will give \(\$ 100\) or more. On average, how many potential donors will she have to contact until she gets her first \(\$ 100\) donor?

Chips Suppose a computer chip manufacturer rejects \(2 \%\) of the chips produced because they fail presale testing. a) What's the probability that the fifth chip you test is the first bad one you find? b) What's the probability you find a bad one within the first 10 you examine?

\- Blood Only \(4 \%\) of people have Type AB blood. a) On average, how many donors must be checked to find someone with Type AB blood? b) What's the probability that there is a Type AB donor among the first 5 people checked? c) What's the probability that the first Type AB donor will be found among the first 6 people? d) What's the probability that we won't find a Type AB donor before the 10 th person?

\- Seatbelts II Police estimate that \(80 \%\) of drivers now wear their seatbelts. They set up a safety roadblock, stopping cars to check for seatbelt use. a) How many cars do they expect to stop before finding a driver whose seatbelt is not buckled? b) What's the probability that the first unbelted driver is in the 6 th car stopped? c) What's the probability that the first 10 drivers are all wearing their seatbelts? d) If they stop 30 cars during the first hour, find the mean and standard deviation of the number of drivers expected to be wearing seatbelts. c) If they stop 120 cars during this safety check, what's the probability they find at least 20 drivers not wearing their seatbelts?

Simulating the model Think about the LeBron James picture search again. You are opening boxes of cereal one at a time looking for his picture, which is in \(20 \%\) of the boxes. You want to know how many boxes you might have to open in order to find LeBron. a) Describe how you would simulate the search for LeBron using random numbers. b) Run at least 30 trials. c) Based on your simulation, estimate the probabilities that you might find your first picture of LeBron in the first box, the second, etc. d) Calculate the actual probability model. e) Compare the distribution of outcomes in your simulation to the probability model.
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