Question

The taste test for PTC (phenylthiocarbamide) is a favorite exercise for every human genetics class. It has been established that a single gene determines the characteristic, and that \(70 \%\) of Americans are "tasters," while \(30 \%\) are "nontasters." Suppose that 20 Americans are randomly chosen and are tested for PTC. a. What is the probability that 17 or more are "tasters"? b. What is the probability that 15 or fewer are "tasters"?

Short answer

(Given that the probability of being a "taster" is 70% and the probability of being a non-taster is 30%)

Step-by-step solution

Using the binomial probability formula

The binomial probability formula is given by: $$P(X=k) = \binom{n}{k} p^k (1-p)^{(n-k)}$$ Here, P(X=k) represents the probability of having k "tasters" in the sample, n is the sample size (in this case, 20), k is the number of "tasters" you want to find the probability for, p is the probability of being a "taster" (0.7), and \((1-p)\) is the probability of being a "non-taster" (0.3).

Part a: Finding the probability of having 17 or more "tasters"

To find the probability of having 17 or more "tasters" in the sample, we need to calculate the sum of the probabilities of having 17, 18, 19, and 20 "tasters". So, we will use the binomial probability formula for k=17, 18, 19, and 20: $$P(X \geq 17) = P(X=17) + P(X=18) + P(X=19) + P(X=20)$$ Calculate each probability using the binomial probability formula and then sum them up to find the total probability.

Part b: Finding the probability of having 15 or fewer "tasters"

Similarly, to find the probability of having 15 or fewer "tasters" in the sample, we need to calculate the sum of the probabilities of having 0 to 15 "tasters". So, we will use the binomial probability formula for k=0, 1, 2, ..., 15: $$P(X \leq 15) = P(X=0) + P(X=1) + ... + P(X=15)$$ Calculate each probability using the binomial probability formula and then sum them up to find the total probability.

Key concepts

Probability Distribution

In statistics, a probability distribution is a function that describes all the possible values and likelihoods that a random variable can take within a given range. If you think of outcomes as different possible results of an experiment, the probability distribution tells us how likely each outcome is to occur.
For example, in a coin toss, the probability distribution is simple since there are only two possibilities: heads or tails, each with a probability of 0.5 under fair coin conditions.

• In our PTC tasting example, the random variable represents the number of tasters among 20 people.
• This follows a binomial distribution because each person can only be a taster or a non-taster, much like a coin flip.
• The formula to determine these probabilities is called the binomial probability formula.

This distribution helps us calculate specific probabilities, such as how many out of 20 individuals are likely to be tasters.

Genetics

Genetics is the scientific study of genes and heredity—how certain qualities or traits are passed from parents to offspring. It explains the genetic variation that exists within individuals or between populations.
In our context with phenylthiocarbamide (PTC) tasting, genetics plays a crucial role in determining if a person can taste PTC or not.

• A single gene with two alleles—one that lets you taste PTC and one that doesn't—determines your ability.
• About 70% of Americans have the allele that allows them to taste PTC.
• These genetic traits follow Mendelian inheritance principles, which are the guidelines for how traits are passed down through generations.

Understanding genetics here helps explain why some people taste certain chemicals while others cannot.

Phenylthiocarbamide (PTC)

Phenylthiocarbamide, often abbreviated as PTC, is a chemical used in taste tests to observe variations in genetic taste sensitivity. An interesting fact about PTC is its bitter taste, which is easily detectable by those with the tasting gene.
PTC testing has a variety of uses, especially in genetics classes and studies, for its role in helping understand how sensory perception can be genetically different among individuals.

• The ability to taste PTC is not directly a health-related trait but serves as a marker for genetic variation.
• This trait is determined by a single gene, making it a classic case study in genetic inheritance.
• Because it is such a simple model, PTC tasting is frequently used in educational settings.

By studying who can taste PTC and who can't, scientists gain insight into genetic diversity and how genes affect perception.

Tasters and Non-Tasters

When it comes to PTC, individuals are broadly categorized into two groups: tasters and non-tasters.
Tasters are those who are genetically predisposed to detect the bitter taste of PTC, while non-tasters cannot perceive it at all. This division in sensory experience is due to a variation in a specific gene.

• It’s estimated that 70% of the American population are tasters, meaning they have at least one copy of the dominant allele responsible for tasting PTC.
• The remaining 30% are non-tasters and generally possess two copies of the recessive allele.
• This categorization is an example of a simple Mendelian trait, illustrating dominance and recessiveness in genetics.

Knowing whether someone is a taster or non-taster helps in studies related to genetics and population variations in taste perception.