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Studentka2010 [4]
2 years ago
5

In a class of 28 students, the teacher selects four people at random to participate in a geography contest. What is the probabil

ity that this group of four students includes at least two of the top three geography students in the class
Mathematics
1 answer:
mixer [17]2 years ago
3 0

Using the hypergeometric distribution, it is found that there is a 0.0452 = 4.52% probability that this group of four students includes at least two of the top three geography students in the class.

The students are chosen without replacement, hence the <em>hypergeometric distribution</em> is used to solve this question.

<h3>What is the hypergeometric distribution formula?</h3>

The <em>formula </em>is:

P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}

C_{n,x} = \frac{n!}{x!(n-x)!}

The parameters are:

  • x is the number of successes.
  • N is the size of the population.
  • n is the size of the sample.
  • k is the total number of desired outcomes.

In this problem:

  • There are 28 students in class, hence N = 28.
  • Four people will be chosen at random, hence n = 4.
  • The top three is composed by 3 people, hence k = 3.

The probability that this group of four students includes at least two of the top three geography students in the class is:

P(X \geq 2) = P(X = 2) + P(X = 3)

In which:

P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}

P(X = 2) = h(2,28,4,3) = \frac{C_{3,2}C_{25,2}}{C_{28,4}} = 0.0440

P(X = 3) = h(3,28,4,3) = \frac{C_{3,3}C_{25,1}}{C_{28,4}} = 0.0012

Then:

P(X \geq 2) = P(X = 2) + P(X = 3) = 0.0440 + 0.0012 = 0.0452

0.0452 = 4.52% probability that this group of four students includes at least two of the top three geography students in the class.

You can learn more about the hypergeometric distribution at brainly.com/question/4818951

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