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There are a total of 12 quotes, half of which were said by President Trump and half by Gordon Gekko.

let X denote a random variable which indicates the number of right answers one has on quotes from President Trump.

Let Y denote a random variable that indicates the number of right answers one has on quotes from Gordon Gekko.

Find the probability of getting a total of 11 correct, P (X + Y = 11)?

It is possible to give an answer based on logic but mathematically I couldn't solve the problem. Furthermore it is possible to see that the probability used is related to hypergeometric.

mustafa badakhsh
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  • Can you clarify your question? It would seem we need to know something about the probability of answering each question correctly, no? – lulu Jul 30 '20 at 19:05
  • Simply put you should find which quotes are said by Trump and which by Gordon. You are provided with information that there exists 6 Trump quotes and 6 Gordon quotes. So the sum of X and Y can be ≤ 12 excluded odd numbers. However I am not able to solve it properly. – mustafa badakhsh Jul 30 '20 at 19:10
  • So...the probability that $X+Y=11$ is? – lulu Jul 30 '20 at 19:14
  • But, for even values you can't hope to answer it since you give us no information at all about the probabilities. Perhaps you always know the right answer, perhaps you have no idea and guess randomly. You get different answers for these cases, as for any other probability you introduce. – lulu Jul 30 '20 at 19:16
  • Should say: depending on how you answer, you can get non-zero values for $X+Y=11$. If you just flip a coin each time, then it is perfectly possible to guess correctly exactly $11$ times. – lulu Jul 30 '20 at 19:20
  • Actually the question is written exactly as I wrote it above. but to clarify furthermore, if you get only one right on Trumps quotes it means that you assigned other five of Trumps quotes as being said by Gordon. And by doing so you have used 5 of Gordons choices on Trump quotes, which means that you can only get one right answer on Gordon's quotes. This in total gives you two correct answers. So you always get even numbers for correct answers. If you get 2 right on Trump you will also get two right on Gordon which gives you 4 correct answers in total. – mustafa badakhsh Jul 30 '20 at 19:23
  • You never said that you choose exactly $6$ Trump quotes. That is critical. – lulu Jul 30 '20 at 19:25
  • so the answer is the probability to get 11 for X+Y is zero. – mustafa badakhsh Jul 30 '20 at 19:25

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If I understand correctly the question - you have 12 quotes, six of which by Trump and you need to pick which ones are by Trump (at random, as you don't recognize any).

In this case, $X$ has a hypergeometric distribution: you pick 6 quotes to attribute to Trump out of a population of 12, which has 6 Trump quotes and want to know how many of the selected ones are by Trump.

Now $Y$. Every quote you wrongly attribute to Trump means there is another Trump quote that you wrongly attribute to Gekko. For example, if $X=6$ then you are correct in all the guesses for Trump and surely will be correct for Gekko. If $X=5$, you missed only one Trump quote which means that in the list of not-chosen quotes, there are 5 correct answers (Gekko) and one wrong (Trump). Conclusion: $Y=X$.

Hence, all probabilities of the sort $\Pr(X+Y=k)$ can be calculated using the Hypergeometric random variable $X\sim HG(12,6,6)$: $$\Pr(X+Y=k)=\Pr(X=\tfrac{k}{2})$$ Clearly, if $k$ is odd, this probability is $0$.

YJT
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