# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-8/5

Exercises: Try Out - Set 8: Real exam simulation to sharpen your skills and build confidence. - Problem 5

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## Exercise 5

### Question

export const metadata = {
    title: "Problem 5",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "12/25/2025",
};

There are two boxes, both containing black and white balls. The number of balls in each box is <InlineMath math="20" />. One ball is taken from each box, and the probability that both balls are black is <InlineMath math="\frac{3}{10}" />. If the probability that both balls are white is <InlineMath math="\frac{m}{n}" />, then <InlineMath math="m + n = \dots" />


### Choices

- [ ] $$3$$
- [x] $$6$$
- [ ] $$7$$
- [ ] $$12$$
- [ ] $$14$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 5",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Let:

- <InlineMath math="P(h_1) =" /> Probability of black in box 1
- <InlineMath math="P(h_2) =" /> Probability of black in box 2
- <InlineMath math="P(p_1) =" /> Probability of white in box 1
- <InlineMath math="P(p_2) =" /> Probability of white in box 2

Given the probability of getting black balls from both boxes is <InlineMath math="\frac{3}{10}" />. Since taking balls from box 1 and box 2 are independent events:

<MathContainer>
  <BlockMath math="\begin{aligned} P(h_1 \cap h_2) &= P(h_1) \times P(h_2) \\ \frac{3}{10} &= \frac{n(h_1)}{20} \times \frac{n(h_2)}{20} \\ \frac{3}{10} &= \frac{n(h_1) \times n(h_2)}{400} \\ n(h_1) \times n(h_2) &= 120 \end{aligned}" />
</MathContainer>

We need to find a pair of factors of <InlineMath math="120" /> where both are less than or equal to <InlineMath math="20" /> (maximum number of balls). The possible pair is <InlineMath math="12" /> and <InlineMath math="10" /> (or vice versa). Let <InlineMath math="n(h_1) = 12" /> and <InlineMath math="n(h_2) = 10" />.

Then the number of white balls is:

- Box 1: <InlineMath math="20 - 12 = 8" /> white balls
- Box 2: <InlineMath math="20 - 10 = 10" /> white balls

The probability that both balls are white is:

<MathContainer>
  <BlockMath math="\begin{aligned} P(p_1 \cap p_2) &= P(p_1) \times P(p_2) \\ &= \frac{8}{20} \times \frac{10}{20} \\ &= \frac{2}{5} \times \frac{1}{2} \\ &= \frac{1}{5} \end{aligned}" />
</MathContainer>

Given <InlineMath math="\frac{m}{n} = \frac{1}{5}" />. So <InlineMath math="m = 1" /> and <InlineMath math="n = 5" />. The value of <InlineMath math="m + n = 1 + 5 = 6" />.


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