# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-9/20

Exercises: Try Out - Set 9: Real exam simulation to sharpen your skills and build confidence. - Problem 20

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## Exercise 20

### Question

export const metadata = {
    title: "Problem 20",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "12/26/2025",
};

In a box, there are <InlineMath math="8" /> yellow balls and <InlineMath math="3" /> blue balls. If two balls are taken simultaneously at random, determine the expected frequency of obtaining two blue balls from <InlineMath math="110" /> trials!


### Choices

- [ ] $$3$$
- [ ] $$5$$
- [x] $$6$$
- [ ] $$9$$
- [ ] $$11$$

### Answer & Explanation

export const metadata = {
  title: "Solution 20",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

First, we calculate the probability of picking two blue balls. The total number of balls in the box is the sum of yellow and blue balls.

<MathContainer>
  <BlockMath math="n(S) = 8 + 3 = 11" />
</MathContainer>

The number of ways to choose <InlineMath math="2" /> balls from a total of <InlineMath math="11" /> balls can be calculated using combinations.

<MathContainer>
  <BlockMath math="C^{11}_2 = \frac{11!}{2!(11-2)!} = \frac{11 \times 10 \times 9!}{2 \times 1 \times 9!} = \frac{110}{2} = 55" />
</MathContainer>

Next, we calculate the number of ways to choose <InlineMath math="2" /> blue balls from the <InlineMath math="3" /> available blue balls.

<MathContainer>
  <BlockMath math="n(A) = C^3_2 = \frac{3!}{2!(3-2)!} = \frac{3 \times 2!}{2! \times 1!} = 3" />
</MathContainer>

The probability of obtaining two blue balls (<InlineMath math="P(2B)" />) is the ratio of the number of ways to pick two blue balls to the total number of ways to pick any two balls.

<MathContainer>
  <BlockMath math="P(2B) = \frac{n(A)}{n(S)} = \frac{3}{55}" />
</MathContainer>

The expected frequency (<InlineMath math="F_h" />) is the product of the probability of the event and the number of trials. The number of trials is <InlineMath math="110" />.

<MathContainer>
  <BlockMath math="F_h = P(2B) \times N" />
  <BlockMath math="F_h = \frac{3}{55} \times 110" />
  <BlockMath math="F_h = 3 \times 2 = 6" />
</MathContainer>

Thus, the expected frequency of obtaining two blue balls is <InlineMath math="6" />.


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