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#51 Re: Help Me ! » Summation help. » 2007-04-05 15:02:10
I found that if I did:
∑(n choose k)*(3^n)*(3^(n-k)) :: (sum from k=0 to n)
Then divided that by 2.
Then divided that by 6^n.
And it gets 50% for all values of n. But not sure why it would work to divide by 2?
Any ideas about that? or the question above? :-)
#52 Help Me ! » Summation help. » 2007-04-05 14:03:43
- clooneyisagenius
- Replies: 7
Problem: When rolling n dice, what is the probability that the sum of the numbers rolled is even.
To answer this I found that:
1. There always has to be an even number of odd dice.
For example, if n=10, there has to be 0,2,4,6,8, or 10 odds for the outcome to be even.
0 odds = (10 choose 0)*3^10
2 odds = (10 choose 2)*3^2*3^8
4 odds = (10 choose 4)*3^4*3^6
6 odds = (10 choose 6)*3^6*3^4
8 odds = (10 choose 8)*3^8*3^2
10 odds = (10 choose 10)*3^10*3^0
Therefore, for any number there must be (n choose k)*(3^n)*(3^(n-k)). but it only works when k=even. Then the number must be divided by 6^n since that's the total number of possibilities there were. And the answer WILL BE 50%.
My question:
How do I find the sum of all of the possible outcomes? I know that if it were for all k (not just evens) it would be:
∑(from k=0 to n) (n choose k)*(3^n)*(3^(n-k)) but I'm not sure how to do it for only k's being even.
Any ideas?
THANKS!
#53 Re: Help Me ! » 3 discrete math problems: 1. countable sets, 2. probability, 3. pascal » 2007-04-04 04:08:43
For #1:
If I assume:
A1={a11,a12,...}
A2={a21,a22,...}
.
.
.
Ak={ak1,ak2,...}
.
.
.
Then there must be a function f(aij) = (j,i) where (j,i) is in NxN (which I have already proven to be bijective. I think I would need to show that f is a bijection too though?
#54 Re: Help Me ! » 3 discrete math problems: 1. countable sets, 2. probability, 3. pascal » 2007-04-03 19:45:32
I agree with you all that the probability is 50% but I was not sure how to prove it exactly. What pi man said looks good to me. I'm not sure if it is exactly what I'm supposed to find; the chapter we are on is about "counting" and "binomial coefficients" and the such. But I think it to be proof enough, and very helpful! Thanks!
I consider #2 done then! 
Thanks!
For #1 I think Stanley_Marsh's response is golden. I understand it completely and actually had most of it. I was unable to get to/past this part:
I'd say I'm about 90% sure on this now though. Still more advice/help is welcome.
Thanks for the help Stanley_Marsh and Pi Man!
#55 Help Me ! » 3 discrete math problems: 1. countable sets, 2. probability, 3. pascal » 2007-04-03 11:42:56
- clooneyisagenius
- Replies: 14
1. Let A1, A2, ... be a sequence of sets, each of which is countable. Prove that the union of all the sets in the sequence is a countable set.
2. When rolling n dice, what is the probability that the sum of the numbers obtained is even. (Note: this does not assume any number of dice in particular. More importantly, not 2 dice.)
3. Use Pascal's Formula to prove by induction on n that (n choose k) = n!/(k!(n-k)!) when 0<=k<=n and (n choose k)=0 otherwise. Assume that (0 choose 0)=1 and (0 choose k)=0 when k is not 0.
Help on any or all of them would be great! I appreciate it! Thanks!
#56 Help Me ! » injective/surjective/bijective help!!!! » 2007-03-25 12:24:36
- clooneyisagenius
- Replies: 2
True or False? Counterexample or Proof?
a) Every decreasing function from R to R is surjective
b) Every nondecreasing function from R to R is injective
c) Every injective function from R to R is monotone
d) Every surjective function from R to R is unbounded
e) Every unbounded function from R to R is surjective
f) f(x) = ax + b is both surjective and injective
My thoughts (and HELP on areas of WHY they are or are not).
a. true, HELP.
b. false, f(x)=x^2 for example.
c. no idea, HELP.
d. true, HELP
e. true, HELP
f) true, HELP.
Thanks so much.