The explanation above is quite net.

What is still not clear ??

Regards

For this type of question, it'd be better to visualize it with a venn diagram as RSQUANT has demonstrated above. It would also help to pick a number of total students at the school, namely one that is a multiple of 3 and 4. This question tests your understanding of overlapping groups/probability.

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If you already understand the concepts, skip to below the italicized for the answer.

For starters, what is the question asking? It's asking for all of the students that are in spanish, OR are left handed, \(OR\) are in both. The OR represents either Spanish student is picked, OR a left handed student is picked, OR a Spanish student AND a left-handed student is picked. The distinction between AND and OR is a very important concept to grasp for word problems such as these.

For this question, using the visual of a venn diagram enclosed in a box, we are focused on everything but the area outside the venn diagram (the neither area). This is RSQUANT's first venn diagram.
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Continuing, we have \(\frac{2}{3}\) of the students can be taking spanish and \(\frac{1}{4}\) can be lefted handed. How do we maximize/minimize the group of both?

I'll be picking 60 as the total number of students. Well, \(\frac{2}{3}\) of 60 is 40 students and \(\frac{1}{4}\) of 60 is 15 students.

So what if ALL of the students that are left handed take spanish?
Then that would mean 15 students are left handed AND take spanish. Notice that this is the both group.

Therefore, in this scenario, the probability of picking a left handed student OR a Spanish student OR both is 40 out of 60 students, since there are 40 students taking spanish, and the 15 left-handed students are subsumed within the group of spanish students (RSQUANTS's third venn diagram).

So we have a minimum of \(\frac{40}{60}\) \(=\) \(\frac{2}{3}\)
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In the next scenario, let's assume that NONE of the left handed students are taking spanish.

That means there is NO overlap between left handed students and spanish students, and in fact, we have some students that are NEITHER left handed or taking spanish.

This is equivalent to both circles in the venn diagram being seperate, as the second venn diagram of RSQUANT shows. Mathematically, this can be seen this way: 40+15 = 55, but we have 60 students total, so those extra 5 students must be right handed and not taking spanish.

Now, how many students take spanish, OR are left-handed, OR take both? We know that's 55 out of 60 students, which is \(\frac{11}{12}\).

\(\frac{11}{12}\) is the maximum.
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So now we have our range: \(\frac{2}{3}\) <= P[S OR L OR (S AND L)] <= \(\frac{11}{12}\).

B,C,D, and E all fall within this range, so they are the answers.

In this question you have to think in the best and worst situation. The best situation is when you have more probabilities and you will have more probabilities if is independent, I mean when there is no a both left-hand and Spanish learning student. In this case you will have a probability of A+B-A*B = 1/4 + 2/3 - 0 = 11/12. The other situation is when you have the lowest probabilities and this is when there are the highest number of both left-hand and Spanish students. In this case the total probabilities will be the the highest number of independent probabilities in this case 2/3. So the probabilities of picking a left-hand or Spanish or both will range from 2/3 to 11/12. B,C,D,E are correct answers.

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