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QUESTION :

Over the last 13 days, there was 33 new stories. It is known that 15 stories are true and 18 are false. When you read the story, you get to chose between 3 choices: You think it is a true story (event A), you think it is a false story (event B) or do research on it (event C).

$P(A)=0.40$ , $P(B)=0.25$ , $P(C)=0.35$

At the moment it is pretty simple. But this is where it gets tricky...

When you decide to do research, here are the probabilities of your next choice depending of the validity of the story.

If the Story is True: $P(Believe)=0.56$ , $P(Rejecting)=0.18$ , $P(Research Again)=0.26$

If the Story is False: $P(Believe)=0.18$ , $P(Rejecting)=0.26$ , $P(Research Again)=0.56$

What is the probability to believe at least 1 false story AND reject 1 true story in 1 day ?

ANSWER :

Since it could be an infinite loop, I guess we have to use some induction here to find the answer.

Lets start with the part where it says we believe at least 1 false and reject 1 true. Is it correct to assume the following equation ?

$M$ : believe a false story , $L$ : reject true story $$P(M>=1 ∩ L>=1)=1-P(M=0 ∪ L=0)$$

So basically I need to reject all false stories OR believe all true stories

EDIT Since i have to reject a true story and believe a true story everyday** for 13 days, and they are spread randomly across those 13 days, so I need the probability to believe at least 13 false stories and reject 13 true stories.

There is 2 things that i don

1. The chances to pick randomly a true story is $15/33=0.45$. But then how is it possible to calculate the probability when you can have an infinite loop (keep researching again and again)

2.I don't understand the part where they say "in 1 day", do you have to do the average stories per day ?

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    $\begingroup$ You can get rid of the infinite loop: For a true story that you originally decide to research, the probability of finally believing it, is $\frac{0.56}{0.56+0.18}$. $\endgroup$ – Hagen von Eitzen Jan 28 '17 at 23:42
  • $\begingroup$ How did u get to that ? Is it because you just don't care if u research again ? you just pick how much of the time you will believe it or not ? $\endgroup$ – Lalous Jan 29 '17 at 0:07
  • $\begingroup$ Since there is $0.40$ chance I believe a story and $0.25$ ill do research, and for a true story I'll believe after research $0.76$(after using your hint) of the time. I can say that I will believe a story $0.40+0.25*0.76=0.59$ in the case of a true story. Is that correct ? $\endgroup$ – Lalous Jan 29 '17 at 0:15
  • $\begingroup$ Yes, your rationale is correct (but use 0.35 probability of further research as you state in the question). What I don't understand is the "in one day" part of the question. There are no rate indications in your description. You just state that there are 33 stories in 13 days. We do not know how these stories arrived/appeared. $\endgroup$ – Thanassis Jan 29 '17 at 2:24
  • $\begingroup$ Oh you are right it is 0.35 :P Exactly, there no other information. They don't explain how the stories are spread over the 13 days...Edited: Actually it said the stories are published randomly over the 13 days $\endgroup$ – Lalous Jan 29 '17 at 2:27

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