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today I was answering a exam and I get a problem which I have no idea how to resolve it. Here is the announcement

$500$ ​​people attend a nightclub. Those who are members of the club pay 14 dlls, and those who are not members paid 20 dlls

All ($100\%$) of those who are members attend, and $70\%$ of non-members attend.

How much money did the club make from these $500$ people?

It does not give more details. Can you give an explanation about how to resolve it?

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If I understood the question correctly, we have no way of knowing how many of the people who attended were members. Did you mean to say that $70\%$ of the people who attended were non-members? In that case, we can find an answer. –  Omnomnomnom Jul 22 '13 at 16:41
    
@Omnomnomnom That's right, there's no way of knowing how many people who attended where members. And for the another question, yes –  Darf Zon Jul 22 '13 at 16:44
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The set of all members certainly is finite with cardinality $n<500$. The complement of this set depends on the universe (such as the set of all humans on Earth); maybe the set of all non-members has cardinality $(7 \text{ billion} - n)$? –  Adriano Jul 22 '13 at 16:44
    
By yes to the other question, do you mean that $70\%$ of people attending were non-members? In that case, we would say that $0.7\times 500=350$ people at the club were non-members. On the other hand, if $70\%$ of the non-members attended, then since we don't know how many non-members exist we can't know how many attended. In general, they can't both be true. –  Omnomnomnom Jul 22 '13 at 16:49
    
The reasoning is the second, when you says: On the another hand, if 70... –  Darf Zon Jul 22 '13 at 16:54
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2 Answers

up vote 2 down vote accepted

Let's start with what we know.

  • The probability that someone attended (denoted by $A$) if they are a member (denoted by $M$) is $P(A | M) = 1$.
  • The probability that someone attended if they are not a member (denoted by $N$) is $P(A | N) = 0.7$
  • #M + 0.7#N = 500, where $\#M$ denotes the number of members.

From the latter, we have that $\#N = \frac{500 - \#M}{0.7}$.

From Bayes' Theorem, we have

$$P(M | A) = \frac{P(A | M)P(M)}{P(A | M)P(M) + P(A | N)P(N)}.$$

This gives us the probability that someone who did attend the party is a member.

Now, we don't know the probability that someone is a member ($P(M)$), or that probability that someone who attended the party is a member ($P(M|A)$). But we can fill some things out and maybe see what we can do.

$$P(M|A) = \frac{1 \cdot P(M)}{1\cdot P(M) + 0.7\cdot P(N)}$$

We can write a similar equation for $P(N|A)$.

Now, the amount of money made is

$$500\left[ 20 P(N|A) + 14 P(M|A)\right].$$

Remember, the probabilities lie in the interval $[0,1]$, which is why we multiply by 500.

So, we have

$$\textrm{Income} = 500 \left[\frac{20 \cdot 0.7 P(N) + 14 P(M)}{P(M)+0.7 P(N)}\right] \\ = 500\cdot 14\left[\frac{P(N)+P(M)}{P(M)+0.7P(N)}\right].$$

But, someone is either a member, or not. So $P(M)+P(N) = 1$.

Now, for the coup de grace.

Remember that $\#N = \frac{500-\#M}{0.7}$. This means that $$P(N) = \frac{\frac{500-\#M}{0.7}}{\#M+\frac{500-\#M}{0.7}}, \\ P(M) = \frac{\#M}{\#M + \frac{500-\#M}{0.7}}. $$

So $$P(M) + 0.7P(N) = \frac{\#M + 500 - \#M}{\#M+\frac{500-\#M}{0.7}} \\ = \frac{500}{\left(1-\frac{1}{0.7}\right)\#M + \frac{500}{0.7}} $$

Therefore, the total money obtained $$\textrm{Income} = 500 \cdot 14 \left[\frac{\left(1-\frac{1}{0.7}\right)\#M + \frac{500}{0.7}}{500}\right]$$ is a function of the total number of members.

If there are no members, $\#M = 0$, then the income is easily found to be $500 \cdot \frac{14}{0.7} = 500 \cdot 20 = 10000$, which matches with the obvious definition.

If there are 500 members, then we have $500\cdot 14\left(\frac{500 - \frac{500}{0.7}+\frac{500}{0.7}}{500}\right) = 500\cdot 14$ pesos income.


As a result, we find that the income is a linear function of the number of total members of the club.

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The assumptions I made here was that the OP did give as complete information as was possible, and that the desired answer was to use probability to derive a relationship between income and membership size. The outcome is exactly the same as if a standard, straightforward approach was taken. I assumed the problem was an exercise in probability to show that the results are equivalent. –  Arkamis Jul 22 '13 at 17:48
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As phrased, the question cannot be answered.

If we knew how much money was made, we'd know how many members attended. If we knew how many members attended, we'd know how much money was made. Since there is no way of determining how many members attended the nightclub, there is no way of finding out how much money was made.

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