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Probability meets Die

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by dtweah » Sun Mar 07, 2010 2:53 pm
A standard, fair, 6-sided die is rolled 8 times. Given that the number
3 appears exactly three times, what is the probability that no two 3's
appear on consecutive rolls?

(a) 5/14
(b) 3/7
(c) 1/2
(d) 4/7
(e) 9/14
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by dmateer25 » Sun Mar 07, 2010 6:02 pm
3 _ 3 _ 3 _ _ _
3 _ 3 _ _ 3 _ _
3 _ 3 _ _ _ 3 _
3 _ 3 _ _ _ _ 3
3 _ _ 3 _ 3 _ _
3 _ _ 3 _ _ 3 _
3 _ _ 3 _ _ _ 3
3 _ _ _ 3 _ 3 _
3 _ _ _ 3 _ _ 3
3 _ _ _ _ 3 _ 3
_ 3 _ 3 _ 3 _ _
_ 3 _ 3 _ _ 3 _
_ 3 _ 3 _ _ _ 3
_ 3 _ _ 3 _ 3 _
_ 3 _ _ 3 _ _ 3
_ 3 _ _ _ 3 _ 3
_ _ 3 _ 3 _ 3 _
_ _ 3 _ 3 _ _ 3
_ _ 3 _ _ 3 _ 3
_ _ _ 3 _ 3 _ 3


So 20 ways for no 3's to appear on consecutive rolls.


The total number is 8C3 = 56


20/56 = 10/28 = 5/14

I will go with A
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by bpgen » Sun Mar 07, 2010 8:16 pm
Well, my take would be D

As it mentioned that number-3 appeared only 3 times, we could determine, how many times consecutive no-3 appeared, it would be 2(in case of hat trick :-)), say P3(2).
And die was rolled 8 times, so 7 possibility for consecutive number(i.e 1st-2nd, 2-3,3-4,4-5,5-6,6-7,7-8)
Hence, P3(2)=2/7

Now there is another chance only one time two no-3 appeared, and third No-3 appeared somewhere else, let say P3(1)
Hence, P3(1)=1/7

Note that, P3(2) and P3(1) are mutually exclusive.

Therefore, probability of all consecutive 3 appeared would be:
P3(2U1)=P3(2) + P3(1) -P3(2 AND1), note P3(2 AND1)=0.
=>2/7+1/7
=3/7

So probability of NO two number-3 occurs is 1-P3(2U1)=4/7
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by dear_xavier » Mon Mar 08, 2010 4:49 am
dmateer25 wrote:3 _ 3 _ 3 _ _ _
3 _ 3 _ _ 3 _ _
3 _ 3 _ _ _ 3 _
3 _ 3 _ _ _ _ 3
3 _ _ 3 _ 3 _ _
3 _ _ 3 _ _ 3 _
3 _ _ 3 _ _ _ 3
3 _ _ _ 3 _ 3 _
3 _ _ _ 3 _ _ 3
3 _ _ _ _ 3 _ 3
_ 3 _ 3 _ 3 _ _
_ 3 _ 3 _ _ 3 _
_ 3 _ 3 _ _ _ 3
_ 3 _ _ 3 _ 3 _
_ 3 _ _ 3 _ _ 3
_ 3 _ _ _ 3 _ 3
_ _ 3 _ 3 _ 3 _
_ _ 3 _ 3 _ _ 3
_ _ 3 _ _ 3 _ 3
_ _ _ 3 _ 3 _ 3


So 20 ways for no 3's to appear on consecutive rolls.


The total number is 8C3 = 56


20/56 = 10/28 = 5/14

I will go with A

If you are brute-forcing it it's much easier and much quicker to instead count the outcomes where at least two 3's are adjacent - 36/56.
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by bpgen » Mon Mar 08, 2010 4:55 am
dtweah, what's the OA and what's the official explanation please?
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