eight dice rolled together. find P that the top numbers add

Question

eight dice rolled together. find P that the top numbers add up to an odd no.

1/8
1/2
1/6Power4
1/6Power6
3Power8/6Power8

this question is from a testprep site - i am confused with their answer, which i'll post later - would like a fresh insight.

Himalayan · Answer

i got 4(3^8) / (6^8)

KillerSquirrel · Answer

My logic is: 

1 dice = 3/6 (there are 3 odd numbers 1,3,5 - so the probability for the outcome to be odd is 3/6)

2 dice = 3/6 - in the first roll we had to get 1,3,5 - in the second roll we will need 2,4,6 in order that the accumulated outcome (i.e e+o = o) will be odd. 

3 dice = 3/6 (numbers 1,3,5)

4 dice = 3/6

and so on... (dice 5 ... dice 8)

3/6 * 3/6 * 3/6 * 3/6 * 3/6 * 3/6 * 3/6 * 3/6 = 3^8/6^8

(edited - see followup post)

hobbit · Answer

nice one... enjoyed the question very much... what is its source?

answer is 1/2

i must admit that this was my initial guess when i read the question... but then i thought on a nice way to reason it.

lets throw 7 dice first. what is the probability of the sum to be odd? for now we don't know so lets say the answer is p (for odd) which means that the probability for even is (1-p).

now lets throw the eighth dice.

what is the probability for odd now?
if the first 7 are odd, then the eighth must be even which gives us 0.5*p
if the first 7 are even, then the eighth must be odd which adds 0.5*(1-p)
summing up we get:

P (for 8 dice) = 0.5p+0.5(1-p) = 0.5*(p+1-p) = 0.5*1 = 0.5 (!!!! regardless of what p was in the first place)

so the answer is 1/2

with the same reasoning you can deduce that p (for 7 dice) is also 1/2...

KillerSquirrel · Answer

hi hobbit - i see your logic and agree, the answer is 1/2 - well reasoned !

qxjmba · Answer

nice reasoning hobbit. 

i did it the longer route, so not as elegant, but quite easy to understand:

ways of getting an even sum total =

all 8 odd = 1 way
all 8 even = 1way
4 odd 4 even = 8c4 ways
2 odd 6 even = 8c6 =8c2 ways
6 odd 2 even = 8c2 ways

probability of each of the above events is (1/2)^8 because p(odd) = p(even) for one die.

thus sum total = (1+1+8c4+2*8c2) /(2^8) = 128/256 = 1/2

javed · Answer

Very well explained

Javed.

Cheers!

ywilfred · Answer

Can be 7 even, 1 odd --> word is eeeeeeeo --> # of ways = 8!/7!1! = 8
Each combination = 1/2^8, so P = 2^3/2^8

Can be 5 even, 3 odd --> word is eeeeeooo --> # of ways = 8!/5!3! = 56
Each combination = 1/2^8, so P = 7(2^3)/2^8

Can be 3 even, 5 odd --> word is eeeooooo --> # of ways = 8!/3!5! = 56
Can be 1 even, 7 odd --> word is eooooooo --> # of ways = 8!/1!7! = 8

Total P = 16(2^3)/2^8 = 2^7/2^8 = 1/2

oops · Answer

yes given answer is 1/2. thanks for the good explanation!

techjanson · Answer

I followed the same logic

eight dice rolled together. find P that the top numbers add

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