# probability calculation

Suppose that in class of $30$ students,there are $17$ girls and $13$ boys. Five are A students from which three are girls. If random student is chosen, what is a probability that student is a girl or an A student.

I did not understand are A students other class? from $30$ students, probability of girl is $17/30$, from A students, probability is $3/5$, but what about the combination of this probabilities?

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## 2 Answers

Well, I guess the A student is the highest rank student. Anyway, let A student be just some subset of these students. We know that there are 3 A girls and 2 A boys. You consider two events: $$E_1 = \{\text{the randomly chosen student is a girl}\}$$ and $$E_2 = \{\text{the randomly chosen student is an A student}\}.$$ So, you have to find the probability of $E_1$ or $E_2$, so $\mathsf P(E_1\cup E_2)$.

Since $E_1$ and $E_2$ are not disjoint ($E_1\cap E_2 = { \text{randomly chosen student is an A girl} }$ ), you have to apply that $$\mathsf P(E_1 \cup E_2) = \mathsf P(E_1)+\mathsf P(E_2) -\mathsf P(E_1\cap E_2) = 1/30(17+5-3) = 19/30.$$

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Hint: We are told that there are five A students, of which three are girls, so two must be boys. You have four groups: 3 A student girls, 2 A student boys, 14 non-A girls, and 11 non-A boys. These are disjoint, so you can add probabilities.

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