Just a few problems I came up with, none of them too hard.
1). 1,2,5,14: find the next term and a formula for the nth term.
2). I have 7 dice, each of which I role 1 time per game. I have to pay $0.50 per game. If the total sum of all my dice is less than 10 or more that 39, I win the jackpot of $1,000,000. Is this game worth playing, assuming I have all the time in the world, will continue playing for ever? What is the expected gain/loss for every time I play?
3). Not really a problem, just something to think about: Is infinity + 1 greater than or equal to infinity?
Originally posted by clandarkfire2) Hell yes you play. The odds of 7d6 being being greater than 39 or less than 10 are pretty good. I simulated 10k trials and won 3 rolls. In fact, this is even a good game if you have to roll either a 7 or a 42, because the odds of rolling either are about 7 in a million, so after a million trials you'd expect to be up about $6.5 Mil. That's a pretty good player advantage.
Just a few problems I came up with, none of them too hard.
1). 1,2,5,14: find the next term and a formula for the nth term.
2). I have 7 dice, each of which I role 1 time per game. I have to pay $0.50 per game. If the total sum of all my dice is less than 11 or more that 38, I win the jackpot of $1,000,000. Is this game worth playing, assumin ...[text shortened]... lly a problem, just something to think about: Is infinity + 1 greater than or equal to infinity?
3) There are only two magnitudes of infinity. Countably infinite, and uncountably infinite.
An example of "countably infinite" is the number of integers, the number of non-negative integers, or the number of even integers. It's easily proven that the magnitude of each of these things are the same.
"Uncountably infinite" would be like the number of real numbers, the number of fractions between 0 and 1, or the number of irrational numbers.
Uncountably infinite is larger magnitude than countably infinite
*Edit*, messed up the bounds on the dice game, re-calculated my sim.
Originally posted by forkedknightFractional numbers, of the type p/q where p and q are integers, and q is not zero, are countable as shown by George Cantor. Real numbers are not.
3) There are only two magnitudes of infinity. Countably infinite, and uncountably infinite.
An example of "countably infinite" is the number of integers, the number of non-negative integers, or the number of even integers. It's easily proven that the magnitude of each of these things are the same.
"Uncountably infinite" would be like the number ...[text shortened]... countably infinite
*Edit*, messed up the bounds on the dice game, re-calculated my sim.
As infinity is not a number, normal laws of arithmetics doesn't apply. But it can be shown that inf+1 is exactly equal to inf itself. Also shown by Cantor.
Originally posted by FabianFnasI think that by "fractions" he meant connected intervals of [0,1]. Or maybe not...
Fractional numbers, of the type p/q where p and q are integers, and q is not zero, are countable as shown by George Cantor. Real numbers are not.
As infinity is not a number, normal laws of arithmetics doesn't apply. But it can be shown that inf+1 is exactly equal to inf itself. Also shown by Cantor.
Originally posted by PalynkaIf he didn't meant fractions as in http://en.wikipedia.org/wiki/Fraction_(mathematics), then he'd used the same formulation that you did.
I think that by "fractions" he meant connected intervals of [0,1]. Or maybe not...
Or maybe not...
Originally posted by FabianFnasI meant exactly what you thought I meant, and I was wrong. The rest of my examples stand.
If he didn't meant fractions as in http://en.wikipedia.org/wiki/Fraction_(mathematics), then he'd used the same formulation that you did.
Or maybe not...
Oh, and for 3) a recursive definition for the nth term of your series is S(0) = 1, S(n) = S(n-1) * 3 - 1
1)
1, 2, 5, 14, 42
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).
2)
There are 6^7 = 279936 possible results.
Of those there are 36 results in which the sum is less than 10.
There are also 36 results in which the sum is more than 39.
Chance of winning is therefore 72/279936 = 1/3888
Expected profit per game: 1/3888 * 1,000,000 + 3887/3888* 0.50 = 257.70
Yes, you should play this game. A lot.
3)
Depends on what subject. In general infinity is not a number, so infinity + 1 does not have any meaning.
In Axiomatic Settheory (for example), Omega represents the 'number' that comes after al the natural numbers. Then, Omega + 1 is larger.
Originally posted by TheMaster37Thank you for using the Online Encyclopedia of Integer Sequences.
1)
1, 2, 5, 14, 42
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).
2)
There are 6^7 = 279936 possible results.
Of those there are 36 results in which the sum is less than 10.
There are also 36 results in which the sum is more than 39.
Chance of winning is therefore 72/279936 = 1/3888
Expected profit per game: 1/3888 * 1 ...[text shortened]... ga represents the 'number' that comes after al the natural numbers. Then, Omega + 1 is larger.
3) could also be
1,2,5,14,41
With S(n) = (3^n + 1)/2
Originally posted by forkedknightIt was just a correction, not a critic.
I meant exactly what you thought I meant, and I was wrong. The rest of my examples stand.
inf+1 = inf
inf*2 = inf
inf*inf = inf
inf^inf = inf
as long as inf = the number of integers, meaning countable infinity.
inf/inf and inf-inf however lacks meaning.
