Ask the Wizard #179 - dice bet

Quote:

Q#1
This looks to be about two players each rolling the dice
and only their roll counts towards their event(s)
in other words, A Race game.
Q#2
The book question is a popular one all over the net and in school.
To me, one or more persons rolls the dice to see which event happens first for that series of dice rolls.

Quote: mustangsally

how to solve the below question (or bet)

Show Spoiler

(I used a Markov chain and a recursion and ran a simulation to get an answer)

Quote: Wizard

However, I was probably afraid at the time I posted the question that just mentioned in Markov Chain would scare off the audience. I tried to show an answer using as simple math as possible.

Quote: mustangsally

"Somebody bet he that he could roll a total of 6 and 8, with two dice, before I could roll a total of seven twice."

Quote: Wizard

Sally, please check if you approve of the new wording.

Quote: Doc

I think that if we are to be adequately nerdy, we would have to ask some questions like "Who gets to start rolling first?"

Quote: Wizard

Sally, please check if you approve of the new wording.

Quote: lilredrooster

when I found out about the correct answer to that puzzle years ago I tried for a long time to figure out if there was some way to take advantage of it while betting on craps in a casino with the bets that they allow.
and I couldn't figure out any way.

Quote: lilredrooster

I don't think there is any way to use it to your advantage. if there was it would be well known. but on the rare chance that one of you guys or gals knows one please let it fly.

Quote: Wizard

Sally, please check if you approve of the new wording.

Quote: Wizard

So the question on the table is the same thing, except the two players take turns rolling and each roll counts for the player rolling only?

Is this wording acceptable:

Players A and B take turns rolling two six-sided dice. Each roll counts towards the goal of the player rolling only. Player A has a goal to roll both a total of six and eight, in either order, and duplicates are allowed. Player B has a goal to roll a total of seven twice. The dice are rolled until either event occurs. If player A rolls first, what is the probability of player A winning? What if player B rolls first?

>    print(end.time - start.time)
Time difference of 4.670225 mins
>  p = count7/sims
>  error = 3.29*sqrt(p*(1-p)/sims)
>  error
[1] 0.0009997058
>  p
[1] 0.4734789
>  p-error
[1] 0.4724792
>  p+error
[1] 0.4744786
>  1-p
[1] 0.5265211
>  ##########################
>  sims
[1] 2700000
>  count68
[1] 1421607
>  count7
[1] 1278393

Quote: Wizard

Players A and B take turns rolling two six-sided dice. Each roll counts towards the goal of the player rolling only. Player A has a goal to roll both a total of six and eight, in either order, and duplicates are allowed. Player B has a goal to roll a total of seven twice. The dice are rolled until either event occurs. If player A rolls first, what is the probability of player A winning? What if player B rolls first?

Quote: Ace2

If player A goes first, the exact answer is 3,955,200 / 6,880,129 =~ 57.487%.

Quote: Wizard

I agree. To prove I solved it too, here are more decimal places: 0.57487294.

If B goes first I get the probability A winning is 0.52671614.

I solved by the same method as Ace.

Quote: Ace2

Let's say the 2 players roll separate sets of dice simultaneously.

Quote: Ace2

What's the probability A wins and what's the probability of a tie?

Quote: Ace2

This is actually a bit easier than the last calc since you don't have to distinguish which player is rolling, so there are only 4 states instead of 8.

Quote: Ace2

C (x1) = 155/216C + 25/216D