Game at local bar
 

It's my grandfathers wake and I'm at a local bar with relatives drinking and thinking about my grandfather's life. There is a dice game going in the bar, a person has four rolls with 5 die for each roll, provided the roller has not already gotten the proper numbers for the game. The roller has to roll 6, 5, 4, consecutively on any roll of the dice then the sum of the last two die is the total for the round. Just to be clear on the rules of the game, an example of the first roll of the round is 6, 4, 2, 2, 2...so the person only gets to keep the 6 and has to roll the other 4 die to try to get a valid roll. On the next roll she rolls a 5, 1, 2, 3...the five is valid, but a 4 would also need to be in the role before the next total could be added, so she basically added a 5 to her total. Now she has 3 die left since the 6 and 5 count as good rolls and are removed from the subsequent roll. At this point the girl rolls a 4, 5, 5, meaning she got the required 6, 5, 4 sequence as well as a very good roll of 10 total with the 5 plus 5. At this point I bet my uncle 50 bucks that no one beats her roll of 6, 5, 4, 10, given the rules I have layed out in this example, of needing the 6, 5, 4 in sequence before the sum of the last 2 die count. I knew I wsa a huge favorite on this bet, although my father did roll an 11 to win the bet. My guess is at the time I was over an 85% fav to win the bet given their were 4 more people to roll and they needed to beat the odds layed out in the example. If more clarification is needed please let me know, but I would really like to know how big of a fav I was when I made the bet.

Um it\'s called Ship, Captain, Crew
 

And your uncle was TRYING to give you $50.

Re: Game at local bar
 

Call the probability that you qualify at all (in the four rolls) P_qualify.

Once you qualify, there is only a 3/36 = 1/12 chance that you can beat 10.

Thus the chance that you beat 10 is:

P_qualify*(1/12)

Which means that you were AT LEAST an 11:1 favorite for this bet if only one more person was rolling. If more than 1 person was rolling, it's slightly less, but you were surely still a huge favorite.

Calculating P_qualify is actually not so easy to do exactly. I'd estimate that it's somewhere between 70 and 90% tho.

HTH,
gm

Re: Game at local bar
 

Gaming Mouse,

I just ran a simulation with 100,000 runs, and only got a 47% chance of qualifying, and a 3.4% chance overall of beating a 10.

-Blank

Re: Game at local bar
 

[ QUOTE ]
I just ran a simulation with 100,000 runs, and only got a 47% chance of qualifying, and a 3.4% chance overall of beating a 10.

[/ QUOTE ]

Nice. I was thinking about writing a simulation too, but was too lazy.

I still think it would be cooler to solve the problem by hand. After thinking about it for 15 or 20 minutes, I couldn't see any good solution. I wonder if there is some clever way to do it, though.

gm

Re: Game at local bar
 

One question... Suppose you roll a 6,5,4 on the first roll and the other two dice are a 5 and 3

Can you keep the 5 and roll a single die 3 times to try and beat your score of 10?

This wasn't clear in the OP.

Re: Game at local bar
 

I had assumed that your score was 8, and you were done.

If that is not true, my other solutions won't be correct.

gm

Re: Game at local bar
 

OK... if you have to stand pat once you get a 6,5,4...

you will score 11 or better only 5.4967% of the time, so with 4 players to roll you are about a 4 to 1 favorite.

The solution was lengthy...
Here's some of the numbers I used...
rolling 5 dice...
7776 possible combinations
3125 contain no 6
2101 contain a 6 but no 5
1320 contain a 6,5 but no 4
1230 contain 6,5,4 but only 50 of these score 11 or 12

rolling 4 dice (after getting a 6)
1296 possible combinations
625 contain no 5
369 contain a 5 but no 4
302 contain 5,4 but only 24 score an 11 or 12

rolling 3 dice
216 possible combinations
125 contain no 4
91 contain a 4 but only 9 score an 11 or 12

Re: Game at local bar
 

[ QUOTE ]
OK... if you have to stand pat once you get a 6,5,4...

you will score 11 or better only 5.4967% of the time, so with 4 players to roll you are about a 4 to 1 favorite.

The solution was lengthy...
Here's some of the numbers I used...
rolling 5 dice...
7776 possible combinations
3125 contain no 6
2101 contain a 6 but no 5
1320 contain a 6,5 but no 4
1230 contain 6,5,4 but only 50 of these score 11 or 12

rolling 4 dice (after getting a 6)
1296 possible combinations
625 contain no 5
369 contain a 5 but no 4
302 contain 5,4 but only 24 score an 11 or 12

rolling 3 dice
216 possible combinations
125 contain no 4
91 contain a 4 but only 9 score an 11 or 12

[/ QUOTE ]

Did you calculate the exact chance that you qualify?

If so, how? Did you just write a program to enumerate it, or did you figure out something clever?

Re: Game at local bar
 

[ QUOTE ]
Did you calculate the exact chance that you qualify?

If so, how? Did you just write a program to enumerate it, or did you figure out something clever?

[/ QUOTE ]
I wrote a program... I'm not that clever... I'm trying now to take the raw data and come up with some formula that will match.

I calculated the chance to qualify to be 77.2426 %