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Have been trying for the last three hours, and going nuts. Please provide HINTS only, not the solution (or answer).

I'm doing this by a dumb approach, way too much of calculations and excel madness. I'm looking for an intelligent shortcut (or many of them). Looking to cut down on things algebraically. Here's the question.

(Dice = plural of die, the thing with the numbers 1 to 6 on its surface.)

Q. 1000 fair and 1000 unfair dice are available in a bag. On a certain day, from 8 a.m. to 6 p.m., 200 dice are randomly selected from the bag every hour and rolled. The faces are recorded and summed, and the rolled dice are then thrown away, ensuring no replacement. The fair dice behave normally ALL the time, but the unfair dice follow a certain pattern:

(a) Between odd and even hours (e.g. between 9 am and 10 am), the unfair dice change their [1, 2, 3] faces to [4, 5, 6] faces for the whole hour. So they become a dice with [4, 5, 6, 4, 5, 6] as the faces.

(b) Between even and odd hours (e.g. between 8 am and 9 am), the unfair dice change their [4, 5, 6] faces to [1, 2, 3] faces for the whole hour. So they become a dice with [1, 2, 3, 1, 2, 3] as the faces.

(c) However, there is an overriding rule: If any of the two bounding hours is prime, then all other unfair dice patterns are discarded, and the unfair dice change ALL their faces to that prime number only. (e.g. [5, 5, 5, 5, 5, 5] from 4 pm to 5 pm and 5 pm to 6 pm.) If both bounding numbers are prime, then the higher one is shown.

Find the probability (correct to 5 decimal places) that the total sum of all throws from 8 am to 6 pm is (a) Even (b) Is a prime

(12 hour format to be followed after 12 noon. 1 o'clock is 1 o'clock, not 13:00 hours).

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    You've got them switched -- "dice" is the plural of "die".2011-08-05
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    Oh yes, how shameful. Corrected. Thank you for pointing out.2011-08-05
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    You write "Between odd and even hours (e.g. between 9 am and 10 am)" -- but it seems that this is in fact the only case in which that rule applies, not just an example?2011-08-05
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    Do you have any reason (mathematical or contextual) to believe that there's a closed form for this? If not, it should best be done by computer.2011-08-05
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    That's how the problem is. If that's the only instance, then it is the only instance. Yes, you are right. And unfortunately no, the source of the problem does not mention is this is better solved by a computer. Which is why I have been trying with Excel as well.2011-08-05
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    @Damien: No, between $10$ and $11$ and between $11$ and $12$ the prime rule applies. Also rule (a) wouldn't apply between $10$ and $11$ even if $11$ weren't prime.2011-08-05
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    @joriki: Correct, and this is what has messed up my calculations a bit.2011-08-05
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    Maybe we can use represent the states of the dice has markov chains between the hours.2011-08-05
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    @Damien: We surely can, but how does that help us calculate the probability for the sum being a prime without using a computer?2011-08-05
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    @Andrew: You've still got some of the singulars and plurals mixed up -- you had some correct "die" singulars in there originally, and you've changed them to incorrect "dice" along with the correct ones.2011-08-05
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    What happens between 2pm and 3pm, when both bounding numbers are prime? // Edit: sorry, just seen that the higher number applies.2011-08-05

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