# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-10/17

Exercises: Try Out - Set 10: Real exam simulation to sharpen your skills and build confidence. - Broken Egg Probability

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## Exercise 17

### Question

export const metadata = {
  title: "Broken Egg Probability",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

A basket contains <InlineMath math="10" /> duck eggs and <InlineMath math="20" /> chicken eggs. Half of the duck eggs are broken, and one-fourth of the chicken eggs are broken.

If one egg is taken from the basket, what is the probability that the taken egg is a duck egg or a broken egg?


### Choices

- [ ] $$\frac{1}{3}$$
- [ ] $$\frac{2}{3}$$
- [x] $$0.5$$
- [ ] $$0.333$$
- [ ] $$\frac{1}{4}$$

### Answer & Explanation

export const metadata = {
  title: "Egg Probability Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Let's define the events as follows:
- <InlineMath math="S" /> = Sample space (total eggs)
- <InlineMath math="D" /> = Event of picking a duck egg
- <InlineMath math="C" /> = Event of picking a chicken egg
- <InlineMath math="B" /> = Event of picking a broken egg

First, we calculate the total number of eggs in the basket:

<MathContainer>
  <BlockMath math="n(S) = n(D) + n(C) = 10 + 20 = 30" />
</MathContainer>

Next, we calculate the number of broken eggs:
- Broken duck eggs: <InlineMath math="\frac{1}{2} \times 10 = 5" />
- Broken chicken eggs: <InlineMath math="\frac{1}{4} \times 20 = 5" />

The total number of broken eggs (<InlineMath math="n(B)" />) is:

<MathContainer>
  <BlockMath math="n(B) = 5 + 5 = 10" />
</MathContainer>

We are asked to find the probability of picking a duck egg **or** a broken egg (<InlineMath math="D \cup B" />). The formula for the probability of the union of two events is:

<MathContainer>
  <BlockMath math="P(D \cup B) = \frac{n(D \cup B)}{n(S)}" />
</MathContainer>

Where <InlineMath math="n(D \cup B)" /> can be calculated using the inclusion-exclusion principle:

<MathContainer>
  <BlockMath math="n(D \cup B) = n(D) + n(B) - n(D \cap B)" />
</MathContainer>

- <InlineMath math="n(D) = 10" /> (number of duck eggs)
- <InlineMath math="n(B) = 10" /> (total number of broken eggs)
- <InlineMath math="n(D \cap B)" /> is the number of eggs that are both duck eggs **and** broken. From the calculation above, there are <InlineMath math="5" /> broken duck eggs.

Thus:

<MathContainer>
  <BlockMath math="n(D \cup B) = 10 + 10 - 5 = 15" />
</MathContainer>

So the probability is:

<MathContainer>
  <BlockMath math="P(D \cup B) = \frac{15}{30} = \frac{1}{2} = 0.5" />
</MathContainer>

Therefore, the probability of picking a duck egg or a broken egg is <InlineMath math="0.5" />.


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