Question

Briggs and King developed the technique of nuclear transplantation, in which the nucleus of a cell from one of the later stages of the development of an embryo is transplanted into a zygote (a single-cell fertilized egg) to see whether the nucleus can support normal development. If the probability that a single transplant from the early gastrula stage will be successful is .65, what is the probability that more than 70 transplants out of 100 will be successful?

Short answer

#Answer# To find the probability that more than 70 transplants out of 100 will be successful, we can use the binomial probability formula. By summing up the probabilities for exactly 71, 72, up to 100 successful transplants, we find the probability of more than 70 successes. This can be calculated as: P(X > 70) = Σ C(100, k) * (0.65)^k * (0.35)^(100-k) for k=71 to k=100 By using a calculator or a programming language like Python, we can calculate the exact probability value.

Step-by-step solution

Identify the variables of the binomial experiment

In a binomial experiment, there are two outcomes (success and failure) in each trial, and the probability of success is the same for each trial. In this case, the number of trials (n) is 100, and the probability of a successful transplant (p) is 0.65. We want to find the probability that more than 70 successful transplants (k > 70) occur.

Use the binomial probability formula

The binomial probability formula is given by: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where P(X=k) is the probability of exactly k successes, C(n, k) is the combination of n items taken k at a time (also written as C(n,k) = n!/(k!(n-k)!)), p is the probability of success, and n is the total number of trials.

Calculate the probability of more than 70 successes

To find the probability of more than 70 successful transplants (k > 70), we need to sum the probabilities of exactly 71, 72, 73, ... up to 100 successful transplants. Mathematically, this can be written as: P(X > 70) = Σ P(X = k) from k=71 to k=100 Which is equivalent to: P(X > 70) = P(X = 71) + P(X = 72) + P(X = 73) + ... + P(X = 100) Using the binomial probability formula, we can calculate this probability as: P(X > 70) = Σ C(100, k) * (0.65)^k * (0.35)^(100-k) for k=71 to k=100

Calculate the sum using a calculator or programming language

Using a calculator or a programming language like Python, we can calculate the sum of all probabilities from k=71 to k=100. This calculation will give us the probability of more than 70 successful transplants out of 100, which is the final answer.

Key concepts

Nuclear Transplantation

Nuclear transplantation, also known as cloning, is a form of asexual reproduction in which the nucleus from a donor cell is injected into an egg cell from which the nucleus has been removed. The resulting cell is genetically identical to the donor nucleus and can develop into a complete organism.

This technique has been especially significant in developmental biology and genetics, as it allows scientists to study the developmental potential of a cell nucleus and the reprogramming of the genetic material. Briggs and King's experiments, as posed in the exercise, were seminal in proving that the nucleus from a mature cell could indeed support the development of an embryo, fundamentally advancing our understanding of cellular differentiation and genetic potential.

For students tackling similar problems, it's crucial to understand that the underlying question of nuclear transplantation is not just a statistical one but also involves grasping complex biological processes. Researching and grasping the biological aspect can greatly enhance the comprehension of the statistical methods applied.

Binomial Experiment

A binomial experiment is a statistical experiment that meets the following criteria: there must be a fixed number of trials, each trial must have only two possible outcomes (commonly referred to as success and failure), the probability of success must be the same for each trial, and the trials must be independent of each other.

In the context of the exercise, treating each nuclear transplantation as a trial with the outcome of 'success' (transplant allows for normal development) or 'failure' (it does not), and assuming each trial is independent and has the same probability of success, creates the framework of a binomial experiment. Understanding the conditions of a binomial experiment is essential for applying the correct statistical formulas and methods, allowing for accurate probability calculations. The idea of independent trials, constant probability, and a binary outcome are aspects that should be well understood to improve students' grasp of the concept and its application in various scenarios.

Combining Function

When discussing probabilities in the context of binomial experiments, the combining function, also known as the 'combination' or simply 'combinatorics', is used to calculate the number of ways in which events can occur.

In the binomial probability formula, the term \( C(n, k) \) represents the combining function, which is the number of combinations of \( n \) items taken \( k \) at a time. The formula \( C(n, k) = \frac{n!}{k!(n-k)!} \) is used here, where \( n! \) denotes the factorial of \( n \) (the product of all positive integers up to \( n \)).

The ability to calculate combinations is vital in determining the probability of a given number of successes in binomial experiments. By summing the probabilities of the desired outcomes, we can find the total probability of an event occurring. Improving one's knowledge of combinations not only assists in solving such statistical problems but also enhances analytical thinking skills valuable in a wide array of mathematical contexts.