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Zorluk: KolayTwo-Way Tables and Probability

A survey of 140 students in the 9th and 10th grades asked about their preferred after-school activity. The results are summarized in the table below.

GradeSportsArtsTotal
9th Grade402060
10th Grade305080
Total7070140

If a student is selected at random from the 9th-grade students, what is the probability that the selected student preferred sports?

  1. A
    27\frac{2}{7}
  2. B
    13\frac{1}{3}
  3. 23\frac{2}{3}Cevap
  4. D
    47\frac{4}{7}

Cevap

23\frac{2}{3}
The correct answer is 23\frac{2}{3}. The question asks for the probability that a student preferred sports, given that the student is in the 9th grade. This means we only consider the 60 students in the 9th grade. Of those 60 students, 40 preferred sports. Therefore, the probability is 4060\frac{40}{60}, which simplifies to 23\frac{2}{3}.

Adım Adım Çözüm

1
Identify the group of interest from which the student is selected.
The selection is made from the 9th-grade students, so the denominator for the probability is the total number of 9th-grade students, which is 6060.
The problem states: 'If a student is selected at random from the 9th-grade students,' establishing this as a conditional probability where the sample space is restricted to 9th graders.
2
Identify the number of students within this group who satisfy the target condition.
The number of 9th-grade students who preferred sports is 4040.
We look at the intersection of the '9th Grade' row and the 'Sports' column in the table.
3
Calculate the probability by dividing the number of target outcomes by the total outcomes in the conditional sample space, and simplify the fraction.
The probability is 4060\frac{40}{60}, which simplifies to 23\frac{2}{3}.
Dividing both the numerator and the denominator by their greatest common divisor, 20, gives the simplified probability.

Anahtar Kavram

Calculating conditional probability from a two-way frequency table by restricting the sample space to a specific row or column.
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Konular

Two-Way Tables and Probability

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