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I'm really good at probability, but this time I seems like I'm not.

My friends asked me a very tricky question, and I want to see if there's anyone who can find out the answer.

Here's the question:

In a survey of 1,000 randomly selected people:

  • 756 believed that the price of the particular product of is reasonable compared to the last year increase,
  • 487 believed that the quality of that product is worse than the last year, and
  • 363 believed both

From here we know:

Product is reasonable + Product is worse = 363

But how about

  • Product is reasonable + Product is not worse
  • Product is not reasonable + Product is worse
  • Product is not reasonable + Product is not worse

I'm trying to figure it out using set like $P(A\cup B\cup C\cup D)$ but seems like hard for me.

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  • $\begingroup$ Please provide explanation how u found the result $\endgroup$ – SomeOne Aug 29 '15 at 18:46
  • $\begingroup$ Your question is phrased in an extremely unclear manner IMHO. The title as well as the first line state probability, but there is no trace of this term for the remaining of the question. In the question itself, the first bullet is extremely confusing. How can you compare a price to an increase??? In your conclusions below, what do you mean Product is reasonable + Product is worse = 363, and how did you get to that? $\endgroup$ – barak manos Aug 29 '15 at 19:08
  • $\begingroup$ I don't know the conventions round here - is the elementary-set-theory tag appropriate here? I would have said it's not. $\endgroup$ – Patrick Stevens Aug 29 '15 at 19:10
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    $\begingroup$ @PatrickStevens I would argue it is as the theory behind how to find the answer deals with De'Morgans laws and inclusion-exclusion and can be visualized via a Venn Diagram, all things which appear in the description of the tag as you hover your mouse over it. I would however argue that the other two tags (statistics and probability-distributions) are not relevant. $\endgroup$ – JMoravitz Aug 29 '15 at 19:14
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For this kind of question, it's often best to make a table. Here's the data when put into a table.

$$ \begin{array}{c|cc|c} &\text{ Is Reasonable } & \text{ Not Reasonable } & \text{ Total} \\ \hline \text{Is Worse} &363 & ??? &487 \\ \text{Is Not Worse} & ??? & ??? & ???\\ \hline \text{Total} &756 & ??? & 1000 \end{array}$$

With the given information, you can fill in all of the question marks.

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