the probability of failing (with grade F) this course is p, [use the answer of question 22 above as the probability, p, here], what is the probability that at most 2 students out of 16 fail this course? {Hint: use binomial distribution} Q26.If the records show that, the probability for a student to get a grade B this course is p, [use the answer of question 19 above as the probability, p, here], what is the probability that exactly 4 students out of 16 will have a grade B for the course? {Hint: use binomial distribution} Q27.What is the probability of selecting a grade A student for the first time either in 2nd or 3rd selection? {Hint: use Geometric distribution with p value as the answer for question 18}
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![**Q25**: If the records show that the probability of failing (with grade F) this course is \( p \), [use the answer of question 22 above as the probability, \( p \), here], what is the probability that at most 2 students out of 16 fail this course? {Hint: use binomial distribution}
**Q26**: If the records show that the probability for a student to get a grade B in this course is \( p \), [use the answer of question 19 above as the probability, \( p \), here], what is the probability that exactly 4 students out of 16 will have a grade B for the course? {Hint: use binomial distribution}
**Q27**: What is the probability of selecting a grade A student for the first time either in 2nd or 3rd selection? {Hint: use Geometric distribution with \( p \) value as the answer for question 18}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1890f4a3-7a27-4daf-8f69-1a7d829b14db%2Fda2bb21f-a954-4252-b4bf-2c7e75fe6322%2Fygec4ob_processed.jpeg&w=3840&q=75)
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