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." \({ }^{\text {" }}\) 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

Answer: a) The probability that 17 or more out of 20 Americans are "tasters" is approximately 0.0478. b) The probability that 15 or fewer out of 20 Americans are "tasters" is approximately 0.6879.

Step-by-step solution

Part a - Probability that 17 or more are "tasters"

In this case, we want to find the probability that 17 or more Americans are tasters in a sample of 20. This means we need to compute the probability of 17, 18, 19, and 20 tasters in the sample. We will use the binomial distribution formula mentioned above: \(P(X \geq 17) = P(X=17) + P(X=18) + P(X=19) + P(X=20)\) So we have: \(P(X \geq 17) = \binom{20}{17}(0.7)^{17}(0.3)^{(20-17)} + \binom{20}{18}(0.7)^{18}(0.3)^{(20-18)} + \binom{20}{19}(0.7)^{19}(0.3)^{(20-19)} + \binom{20}{20}(0.7)^{20}(0.3)^{(20-20)}\) Calculating these values, we get: \(P(X \geq 17) \approx 0.0478\)

Part b - Probability that 15 or fewer are "tasters"

In this case, we want to find the probability that 15 or fewer Americans are tasters in a sample of 20. This means we need to compute the probability of 0 to 15 tasters in the sample. We will again use the binomial distribution formula: \(P(X \leq 15) = \sum_{k=0}^{15} P(X=k) = \sum_{k=0}^{15} \binom{20}{k}(0.7)^{k}(0.3)^{(20-k)}\) Computing this sum, we get: \(P(X \leq 15) \approx 0.6879\) To summarize: a. The probability that 17 or more out of 20 Americans are "tasters" is approximately \(0.0478\). b. The probability that 15 or fewer out of 20 Americans are "tasters" is approximately \(0.6879\).

Key concepts

Probability Theory

Probability theory is the mathematical framework for understanding randomness and making quantitative predictions about events whose outcomes cannot be determined with certainty. At its core, it offers a set of principles and laws, which help evaluate the likelihood of occurrences within a specific set of parameters. For instance, when flipping a fair coin, probability theory provides the tools to predict the chances of landing heads or tails—each event having a probability of 0.5 or 50%.
In the context of genetics, such as the PTC taste test, probability theory allows us to calculate how likely it is for a randomly chosen individual to have a certain genetic trait, provided we know the overall distribution of that trait in the population. Understanding this theory is fundamental for scientists and statisticians to interpret data and formulate conclusions about genetic predispositions and other traits in populations.

Binomial Theorem

The binomial theorem is a formula for expressing the powers of sums. Specifically, it allows us to expand expressions of the form \( (a+b)^n \) into a sum involving terms of the form \( a^k b^{n-k} \) multiplied by a combinatorial coefficient \( \binom{n}{k} \)—this coefficient represents the number of distinct ways to choose \( k \) elements out of \( n \) total elements. In probability theory, the binomial theorem plays a crucial role in deriving the binomial distribution, which is used to describe the number of successes in a sequence of independent experiments.
When calculating the probability of 'tasters' in the PTC taste test scenario, the binomial theorem helps to determine the likelihood of a certain number of people tasting PTC out of a larger group, provided the probability of any one person being a taster or non-taster is known.

Genetics and Probability

Genetics and probability are intertwined in that the inheritance and manifestation of genes can be understood and predicted using the laws of probability. This connection forms part of the foundation of the study of heredity and genetic variation within populations. For example, if a trait is known to be controlled by a single gene with two alleles, the probabilities of the offspring inheriting particular combinations of alleles can be calculated using probability theory.
This applies to the taste for PTC, a bitter compound. The ability to taste PTC is determined by a single gene with two alleles - tasting (T) and non-tasting (t). With simple Mendelian genetics, one can predict the likelihood of an individual being a taster or a non-taster. In a population, if the proportion of tasters and non-tasters is known, these probabilities can guide predictions about the distribution of these traits in a random sample of individuals.

PTC Taste Test

The PTC taste test is an example of a simple genetic test to determine whether a person has the genetic ability to taste phenylthiocarbamide, a bitter chemical that only some individuals can detect. This characteristic serves as a classic example of a trait that follows Mendelian genetics - being controlled by a single gene with dominant (taster) and recessive (non-taster) alleles. The frequency of 'tasters' and 'non-tasters' within a population can vary, but across broader demographics, general percentages can be established as observed in the American population.
The test's simplicity and clear Mendelian pattern make it an excellent candidate for applying the binomial distribution in probability theory. By assuming that the trait follows a binomial distribution, educators and students can use the known percentage of the taster phenotype in a population to calculate the probability of encountering a certain number of tasters in a randomly selected group, thus reinforcing the concepts of both binomial distribution and genetics.