My wife and I go back and forth on this one. Assume that you already have your small straight. Lets say your 5 dice are 1,2,3,3,4. To get a large straight, should you re-roll the 3, or re-roll the 1 AND 3? My wife says just the 3, because that way you only need 1 die to go your way instead of 2. I say the 1 AND 3 because it doubles your chances of getting a 5 which you need, and you still have a 1/3 chance that the second dice is a 1 or 6.
Looking for some probability experts to help me through this one.
In: Mathematics
Let’s do the math !
First launch :
1,2,3,3,4
Your wife plays. She launches the 3. Her strategy is : “I launch this die until I get a five !”
1. She gets a 5 (1/6).
2. She does not (5/6). She launches again, and gets it (1/6).
So she has 1/6+5/6*1/6=1/6+5/36=0.3055…
Now, you. The strategy is : “Launch the 1 and the 3 until you get a 5 (you store it) or a 1-6 (you store it)”.
1,2,3,3,4
You take the 1 and 3 and launch them. You need a 5, and either a 1 or a 6.
So, what can happen if you plan on winning this :
1. The first dice is a 5 (1/6), and
1. the second dice is a 1 or a 6 (1/3)
2. the second dice is neither a 1 or a 6 (2/3), but you throw it again and it’s one (1/3)
2. The first dice is a 1 or a 6 (1/3), and
1. the second dice is a 5 (1/6)
2. it’s not (5/6), but you launch it again and you get it (1/6)
3. The first dice is neither a 5, a 1 or a 6 (1/2), and
1. the second is a 5 (1/6). You launch the first one again, and it’s a a 1 or a 6 (1/3)
2. the second is a 1 or a 6 (1/3). You launch the first one again, and it’s a 5 (1/6)
3. the second is neither a 5, a 1 or a 6 (1/2). You launch the two of them again and
1. the first one is a 5 (1/6), the second one a 1 or a 6 (2/3)
2. the exact opposite
So, we get :
1/6*(1/3+2/3*1/3)+1/3*(1/6+5/6*1/6)+1/2*(1/6*1/3+1/3*1/6+1/2*(1/6*2/3*2))=0.305…
Well. It’s the same thing, exactly. (And TIL, too, I was instictively going for your strategy).
My old math teacher used to tell me that when I get twice the same result, I should try to understand the why, because there should be a shortcut in the reasoning. Here I just bruteforced it. But it’s beer-time with my roomate, I’ll edit later if I find the courage to dig into it further.
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