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Thank you
NIck
I GOTTA be missing something. Or am I?
---
The odds of any 7 six times in a row seems to be 85.7% ^ 6 = 39.7%
The odds of hitting a 12 once in 6 tries seems to be 14.3% * 6 = 85.7%
The obvious problem is that it adds up to more than 100%
Back to the drawing board....
That's where my error is.Quote: DJTeddyBearThe odds of hitting a 12 once in 6 tries seems to be 14.3% * 6 = 85.7%
Since the problem is resolved prior to six throws once a 12 shows up in the first 5 throws, you have to either create a very complex formula that takes that into account, or calculate for the event that requires all six throws and subtract from 100%
I.E. The odds of hitting six 7s in a row, an event that requires all six throws to resolve, is ( 6 / 7 ) ^ 6 = 39.7%
Therefore the odds of NOT doing that, i.e. hitting AT LEAST one 12 in those six throws, is 100% - 39.7% = 60.3%
THEREFORE: Bet on 12.
Quote: nickMy question is based on dice odds. I know that there are 6 ways to get 7 and 1 way to get 12 but what are the chances of getting 6 sevens before 1 twelve. Are they even and if not how many 12s should be added to the equation to make it an even proposition?
Thank you
NIck
Interesting question. I didn't feel like tacking it direcctly, because a simple simulator was about a two minute programming project. Here are the results of a simulation of one billion (1,000,000,000) rolls of the dice for four separate cases.
Here are the simulation results for four 7's vs. one 12 --
Number of times four 7's came up before a 12: 32581424
Number of times a 12 came up before four 7's: 27768376
Here are the simulation results for nine 7's vs. two 12's --
Number of times nine 7's came up before two 12's: 13442654
Number of times two 12's came up before nine 7's: 10107307
Here are the simulation results for five 7's vs. one 12 --
Number of times five 7's came up before a 12: 23914613
Number of times a 12 came up before five 7's: 27781426
Here are the simulation results for six 7's vs. one 12 --
Number of times six 7's came up before a 12: 18256704
Number of times a 12 came up before six 7's: 27781741
Here is the sloppy C code for the latter of these four simulations. I did not audit or double check this code:
#include
#include
#include
main() {
int d1, d2, s; // dice and dice sum
int noS = 0, noT = 0; // counters for sevens and twelves
int ctr, x = 0, y = 0; // other counters
srand(time(NULL));
x = 0;
y = 0;
for (ctr = 0; ctr < 1000000000; ctr++) {
if (ctr%1000000 == 0)
fprintf(stderr, "%d\r", ctr);
d1 = rand()%6+1;
d2 = rand()%6+1;
s = d1+d2;
if (s == 7)
x++;
if (s == 12)
y++;
if (x == 6) {
noS++;
x = 0;
y = 0;
}
if (y == 1) {
noT++;
x = 0;
y = 0;
}
}
printf("\n");
printf("Number of times six 7's came up before a 12: %d\n", noS);
printf("Number of times a 12 came up before six 7's: %d\n", noT);
}
--Dorothy
Quote: nick... how many 12s should be added to the equation to make it an even proposition?
I've played around with my program to answer this more formal version of your question: Assume the probability of getting X sevens before Y twelves is approximately 0.5. Solve for integer values of X and Y.
I tried values through Y = 6 and was not able to get an implied integer solution for X.
For example, this code examines the problem with X = 34 and Y = 6, 100M rounds:
#include
#include
#include
main() {
int s, d1, d2;
int noS = 0, noT = 0;
int ctr, x = 0, y = 0;
srand(time(NULL));
x = 0;
y = 0;
for (ctr = 0; ctr < 100000000; ctr++) {
if (ctr%1000000 == 0)
fprintf(stderr, "%d\r", ctr);
d1 = rand()%6+1;
d2 = rand()%6+1;
s = d1+d2;
if (s == 7)
x++;
if (s == 12)
y++;
// input value of x 7's before y 12's
if (x == 34) {
noS++;
x = 0;
y = 0;
}
// input value of y 12's before x 7's
if (y == 6) {
noT++;
x = 0;
y = 0;
}
}
printf("\n");
printf("Number of times 7's completed first: %d\n", noS);
printf("Number of times 12's completed first: %d\n", noT);
}
->OUTPUT->
Number of times 7's completed first: 294860
Number of times 12's completed first: 283584
--Dorothy
nick
Quote: nickBasically you are agreeing that it is approximately even odds whether you get 6 consecutive 7s or 1 12 before those 6 7s are thrown
I'm not saying this in the least. Here are the results of my simulation in this case, as posted above:
Number of times six 7's came up before a 12: 18256704
Number of times a 12 came up before six 7's: 27781741
You can see that it is much more likely that a 12 will be rolled than rolling six 7's.
Also, in your original post you did not say "consecutive." Here is the text of your question:
Quote: nickwhat are the chances of getting 6 sevens before 1 twelve.
You simply asked for the probability of rolling six 7's before a 12 was rolled.
Using the numbers from my simulation, the probabilities are about:
Probability of rolling six 7's before a 12 is rolled: 39.655%
Probability of rolling a 12 before rolling six 7's are rolled: 60.345%
This isn't close to 50%.
Again, I am not agreeing with you in any respect here.
--Dorothy
The probability of rolling a 7 is 1/6, and the probability of rolling a 12 is 1/36. The probability of rolling a 7, given that a roll is a 7 or 12 is (1/6)/((1/6)+(1/36)) = 6/7. So the probability that the first six times a 6 or 12 is rolled it is a 6 every time is (6/7)^6 = 39.66%.
If you rephrase the question to be what is the probability of rolling five sixes before a 12, then the answer is (6/7)^5 = 46.27%. With four rolls it is (6/7)^4 = 53.98%. So there is no number that is exactly 50/50. If you're looking for a good sucker bet, suggest you can either roll four sevens before a twelve, or a twelve before five sevens.
Quote: DJTeddyBearThe odds of hitting six 7s in a row, an event that requires all six throws to resolve, is ( 6 / 7 ) ^ 6 = 39.7%
Therefore the odds of NOT doing that, i.e. hitting AT LEAST one 12 in those six throws, is 100% - 39.7% = 60.3%
I underlined and fixed your typos.Quote: Wizard...the probability that the first six times a 7 or 12 is rolled it is a 7 every time is (6/7)^6 = 39.66%
Woo hoo! I DID have the right answer (eventually).
Thanks for the confirmation, Wiz!
I really enjoyed stretching out my brain muscle on this one. :)
Quote: nickMy question is based on dice odds. I know that there are 6 ways to get 7 and 1 way to get 12 but what are the chances of getting 6 sevens before 1 twelve. Are they even and if not how many 12s should be added to the equation to make it an even proposition?
Thank you
NIck
Simplify your equation to a seven sided dice with 1 12 and 6 sevens.
The odds of rolling 6 7s before a 12 is (6/7)^6 = 39.657%.
The odds of rolling 5 7s before a 12 is (6/7)^5 = 46.266%
The odds of rolling 4 7s before a 12 is (4/7)^4 = 53.977%
nick
noQuote: nickthat if I upped the number to 9 sevens vs. 2 twelves it would be an even bet
it would not be an even bet
I have this for
nine 7s B4 two 12s: 0.570822175
two 12s B4 nine 7s: 0.429177825
average number of rolls to either: 42.46
to get closer, maybe, we need to go to
ten 7s and two 12s
ten 7s B4 two 12s: 0.519855909
two 12s B4 ten 7s: 0.480144091
average number of rolls to either: 45.58
Markov chain was used
might be possible to extend this using more 7s...
looking to see if this could be done another way (maybe in a calculator)
not much found about this type of question
for a general solution
Sally
You can do this problem with an algebraic formula. the method (and formula) is a Poisson processesQuote: pacomartinUnfortunately I don't have time to figure that out right now. However, you must set up a Markov transition matrix to cover the situation. You can't do this problem with an algebraic formula. It's much more complex than that.
that can be found in the book "Introduction to Probability Models Ninth Edition" by Sheldon M. Ross
on page 316.
This is true IF and only IF one is after an event before another event or events and NOT an event and another event (like the popular 2 dice roll a 6 and 8 before two 7s question and bar bet found here: https://wizardofodds.com/ask-the-wizard/179/)
*****
these type of questions can be seen as a 'race' between two independent events
in other words, which 'event' happens first in a series of trials.
I changed the formula a little to using the probabilities instead of the mean time for dice examples only
R code here and
online section 1r.
https://sites.google.com/view/krapstuff/dice/dice-probability-math
headsBeforeTails <- function(heads,pheads,tails,ptails){
pTotal <- pheads+ptails
k <- heads:(heads+tails-1)
prob <- sum(choose(heads+tails-1,k)*(pheads/pTotal)^k *(ptails/pTotal)^(heads+tails-1-k))
print(formatC(prob, digits=8),quote=FALSE)
print(formatC(1-prob, digits=8),quote=FALSE)
}
> headsBeforeTails(2,10/36,2,6/36)# two 6s or 8s before two 7s
[1] 0.68359375
[1] 0.31640625
as can be seen a different question and result from "roll a 6 and 8 before two 7s"
the OP in this thread can easily be solved using the formula (and a Markov chain also, but that does require a little more time)
reviewQuote: nickif I upped the number to 9 sevens vs. 2 twelves it would be an even bet;
and to extend the logic even further, and I'm not sure it would hold, would it also be an even bet if I substituted one 11 for two 12s since there are two ways to make 11 and only one way to make 12?
And if I used the number 10 , would the math be (6/9)(6/9) = close to a two to one bet?.
> headsBeforeTails(9,6/36,2,1/36)# nine 7s before 2 twelves
[1] 0.57082217
[1] 0.42917783
> headsBeforeTails(9,6/36,1,2/36)# nine 7s before 1 eleven
[1] 0.075084686
[1] 0.92491531
> headsBeforeTails(9,6/36,1,3/36)# nine 7s before 1 ten
[1] 0.026012295
[1] 0.97398771
well,Quote: nickIt looks like two 12's and eight 7's would be the answer; but that's like one 12 and four 7's.
> headsBeforeTails(8,6/36,2,1/36)# eight 7s before 2 twelves
[1] 0.62433675
[1] 0.37566325
> headsBeforeTails(6,6/36,1,1/36)# six 7s before 1 twelve
[1] 0.39656946
[1] 0.60343054
> headsBeforeTails(5,6/36,1,1/36)# five 7s before 1 twelve
[1] 0.46266437
[1] 0.53733563
> headsBeforeTails(4,6/36,1,1/36)# four 7s before 1 twelve
[1] 0.53977509
[1] 0.46022491
> headsBeforeTails(3,6/36,1,1/36)# three 7s before 1 twelve
[1] 0.62973761
[1] 0.37026239
should be just thinking about it.Quote: nickDo you think there's a different dynamic with more numbers ?
of course probability can throw one a 'curve ball' just when we think we have thought it out!
how about X {4,5,6,8,9 or 10} before two 7s?
> headsBeforeTails(8,24/36,2,6/36)#place{4,5,6,8,9 or 10} B4 two 7s
[1] 0.43620762
[1] 0.56379238
> headsBeforeTails(7,24/36,2,6/36)#place B4
[1] 0.50331648
[1] 0.49668352
> headsBeforeTails(6,24/36,2,6/36)#place B4
[1] 0.5767168
[1] 0.4232832
> headsBeforeTails(5,24/36,2,6/36)#place B4
[1] 0.65536
[1] 0.34464
> headsBeforeTails(4,24/36,2,6/36)#place B4
[1] 0.73728
[1] 0.26272
> headsBeforeTails(3,24/36,2,6/36)#place B4
[1] 0.8192
[1] 0.1808
> headsBeforeTails(2,24/36,2,6/36)#place B4
[1] 0.896
[1] 0.104
the choices look endless (thinking about more dice)
Sally
added:
(like the popular 2 dice roll a 6 and 8 before two 7s question and bar bet found here: https://wizardofodds.com/ask-the-wizard/179/)
this can be solved using a different formula (requires calculus)adjusting one from here:
https://wizardofvegas.com/forum/questions-and-answers/math/22096-one-die-question/5/
a Markov chain or brute force (like the Wizard did enumerating all the possible winning outcomes... a time breaker for a 'race' with more events.)
another day for that!
Quote: mustangsallyYou can do this problem with an algebraic formula. the method (and formula) is a Poisson processes
that can be found in the book "Introduction to Probability Models Ninth Edition" by Sheldon M. Ross on page 316.
Link to the 10th edition