# Roll the dice

#### aloha

Player A and Player B roll a dice. They roll 5 times each. Who dices "5" first will win the game? If player A plays first, what is prob of Player A winning the game. Is it unfair?

#### alexagak

If A plays first and the one who rolls a 5 first wins: If A wins, B gets fewer rolls than A. If B wins, A gets the same number of rolls as B. So the game is unfair (in favor of A) and [prob A winning]>[prob B winning]. There is also the possibility that there is no winner within these 5 rounds of rolls.

• roll 1: win A= 1/6 win B= (5/6)*(1/6) no win= 1-(11/36)
• roll 2: win A= (1/6)*(25/36) win B= (5/6)*(1/6)*(25/36) no win= 1-(275/1296)
• roll 3: win A= (1/6)*(1021/1296) win B= (5/6)*(1/6)*(1021/1296) no win= 1-(11231/46656)
• roll 4: win A= (1/6)*(35425/46656) win B= (5/6)*(1/6)*(35425/46656) no win= 1-(389675/1679616)
• roll 5: win A= (1/6)*(1289941/1679616) win B= (5/6)*(1/6)*(1289941/1679616) no win= 1-(14189351/60466176)

Totals:
1. win A= (40406838/60466176)/5= 0.1336510448419956307473454249860285525580450134633948
2. win B= (33672365/60466176)/5= 0.1113758707016630256227878541550237937983708445528290
3. no win= (168361825/60466176)/5= 0.7549730844563413436298667208589476536435841419837761

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#### Plato

MHF Helper
Player A and Player B roll a dice. They roll 5 times each. Who dices "5" first will win the game? If player A plays first, what is prob of Player A winning the game. Is it unfair?
Reply #2 many well be correct. But I cannot read it. That is clearly evidence that serious helpers should know LaTeX.

Let $$\displaystyle X$$ be the number of rolls on which the game is over.
Note that $$\displaystyle X=1,2,\cdots,10$$.

A wins if $$\displaystyle X=1,3,5,7,9$$ and B wins if $$\displaystyle X=2,4,6,8,10$$

So $$\displaystyle \mathcal{P}(X=k)=\left( {\frac{1}{6}} \right)\left( {\frac{5}{6}} \right)^{ k-1}$$.

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• 1 person

#### alexagak

I thank Plato for the positive criticism.

I would also like to add that reply #3 points out the method, while mine (reply #2) provides the results.

#### Plato

MHF Helper
I would also like to add that reply #3 points out the method, while mine (reply #2) provides the results.
That is exactly the point. It is the policy of this forum not to give complete results. We are not a homework service.

#### alexagak

Ok then, my bad for not going through the policies.

I cannot see how you explicitly made that point in reply #3 as you claim but I will refrain from similar practices in the future.

#### Plato

MHF Helper
I cannot see how you explicitly made that point in reply #3 as you claim.
My my reply, I showed what had to be done. I did not do it.

I left a good bit of work to be done.

#### alexagak

My my reply, I showed what had to be done. I did not do it.

I left a good bit of work to be done.
Indeed you left out the calculations.

I am referring to the fact that we could have skipped this meaningless exchange of replies if you had just pointed out the policy violation from the beginning instead of arguing LaTeX.

In reply #3 you made a point about text readability, not solution-provision policy. And thank you for deciding on whether I know LaTeX or not from my first ever reply on this forum.

#### Plato

MHF Helper
In reply #3 you made a point about text readability, not solution-provision policy. And thank you for deciding on whether I know LaTeX or not from my first ever reply on this forum.
I am glad that you may know LaTeX. I look forward to your using it in the you replies. We are encouraging all serious helpers to use it. That was the real purpose of my comment.