https://file.aiquickdraw.com/mathbot/file/1725975828006t4jh2pn6.jpg

Let's break down and solve the problem presented in the image.

### Part (a):
The problem asks to state four assumptions required to model the number of people with this genetic marker in a sample of size \(n\) as a binomial distribution.

To model a problem using the binomial distribution, the following assumptions must be met:

1. **Fixed number of trials**: The number of trials, \(n\), is fixed. In this case, the sample size of the population is fixed (for instance, a sample of 50).
   
2. **Two possible outcomes**: Each trial has only two possible outcomes: either a person has the genetic marker or they do not. These outcomes are typically called "success" and "failure."

3. **Constant probability of success**: The probability of success (having the genetic marker) is the same for each trial. Here, the probability of a person having the marker is given as 4% (i.e., \(p = 0.04\)).

4. **Independent trials**: Each trial is independent of the others. This means whether or not one person has the genetic marker does not affect whether another person has it.

### Part (b):
Now, the problem asks to find the probability of exactly 6 people having the marker in a sample of size 50.

Since this is a binomial distribution problem, the probability mass function for a binomial distribution is:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

Where:
- \(n = 50\) (sample size),
- \(k = 6\) (we are interested in the probability of exactly 6 people having the marker),
- \(p = 0.04\) (probability of success, i.e., having the marker).

Let's compute this probability. I'll calculate the value for you.The probability of exactly 6 people having the genetic marker in a sample of size 50 is approximately \( 0.0108 \), or 1.08%.

Would you like further clarification on this, or do you have any other questions?

Here are some related questions to extend your understanding:

1. How would the probability change if the sample size increased to 100?
2. What is the expected number of people with the genetic marker in a sample of size 50?
3. How does the binomial distribution differ from the normal distribution in this context?
4. What is the probability of at least 6 people having the genetic marker in the same sample size?
5. How would we model this situation using a Poisson distribution approximation?

**Tip**: The binomial distribution can be approximated by a normal distribution if \(n\) is large and \(p\) is neither too small nor too large!

A particular genetic marker is present in 4% of the population.
(a) State four assumptions that are required to model the number of people with this genetic marker in a sample of size n as a binomial distribution.
(b) Using this model, find the probability of exactly 6 people having this marker in a sample of size 50.

Learn how to calculate the probability of exactly 6 people having a genetic marker in a sample of 50 using the binomial distribution. This problem covers key probability concepts and binomial distribution formulas, suitable for students in Grades 10-12.

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