ugrant:
When you presume more than one condition and apply the presumptions to simple random odds, the combined conditions mean that the (separate) odds multiply together.
Example: the question, "what are the odds of flipping heads?" is one condition, and the answer is 0.5; but the question, "given that I flip a coin exactly two times, what are the odds of flipping heads both times?" the answer is 0.25 (which is 0.5 times 0.5), even though the odds for each flip is 0.5.
Why? Because the question PRESUMES exactly two flips (a condition), and asks the odds of getting heads on both (another condition); the conditions you presume mean that the odds (0.5) are multiplied together (0.5 times 0.5 equals 0.25).
In your question, the presumptions and conditions are:
IF
1) The second batter has a different best column from the first batter (because you've deliberately designed your line-up that way),
AND
2) the first batter gets a roll on his best column (because that's a condition of your question),
THEN
what are the odds that the second roll will be
BOTH (repeat: BOTH)
a) on a different column
AND
b) that column will be the second batter's best column?
The odds of "a" are 5/6ths (0.83); the odds of "b" are 1/5th (0.2) (because of the first "if" above). Combining BOTH these requirements creates 5/6 times 1/5, which equals 1/6.
You are arguing "But what if we presume the first roll lands on the first batter's best column, and the second lands on a different column? Isn't the only relevant probability that of 'b,' giving the answer of 0.2?"
No, because the only reason that the "0.2" appears in the equation is that you have ALREADY stipulated (in condition "a") that the second roll is on a different column. The "0.2" would not appear at all without condition "a".
It's like asking, "What are the odds of flipping a coin twice and having it come up heads twice in a row, IF THE FIRST TOSS IS HEADS?" The answer would be 0.5, but only because you have already stipulated the first toss. The answer without that stipulation is "0.25," which is 0.5 times 0.5.
The same goes for your question. The answer is 0.83 times 0.2, not simply 0.2. The number 0.2 is relevant only because you have already stipulated that the first and second rolls land on different columns, just as in my example it stipulates that the first toss is heads.
To sum up: You got the figure "0.2" because you started out by STIPULATING that figure. It's a prior condition, not a result. You started with a presumption, and concluded with your own presumption. You set the terms of the argument, and then answered the argument with the terms you set. That's tautology, not statistics.
The secret to success
by ClowntimeIsOver » Mon Sep 14, 2009 9:25 pm
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by ugrant » Tue Sep 15, 2009 11:42 am
maligned: you wrote: "You lay out perfectly a strategy for gaining the 20% to 16.7% advantage directly after a previous successful roll [b:ba1ba72fd7]in an environment where repeat rolls are not possible[/b:ba1ba72fd7]."
I'm not following this, except to say nice use of bolding to make your point. In my last post scenario A clearly assigns the correct value of 1/6 to a repeat roll occurring. Combined with scenario B, all possible outcomes of a roll of a six sided die are covered - 1/6 the die repeats and 5/6 it does not. Somebody's missing something, I guess.
wachovia: I don't remember saying anything about presumptions or conditions, I've always just quoted odds and how they apply to a die roll. I read your post twice and actually started answering each of your assertions before coming to the conclusion I don't get how flipping a coin applies to a die roll (guess that puts me in the company of those folks who didn't get how the Monty Hall game show scenario applies to die rolls). Sorry. Maybe you've identified how superior you might be in the intelligence department (and, yes, I know that just threw a softball to those of you who have been questioning my intelligence).
Tautology? Wow - a true Verbal Advantage word. I had to look it up. From Wikipedia:
Tautology (rhetoric), repetition of meaning, using dissimilar words to say the same thing twice, especially where the additional words fail to provide additional clarity and meaning.
Tautology (logic), a technical notion in formal logic, universal unconditioned truth, always valid.
If you're applying either of those to me, I think you give me far too much credit.
Very entertaining clip, Turtle. Rmilter would love that with his entirely same column lineup.
I'm not following this, except to say nice use of bolding to make your point. In my last post scenario A clearly assigns the correct value of 1/6 to a repeat roll occurring. Combined with scenario B, all possible outcomes of a roll of a six sided die are covered - 1/6 the die repeats and 5/6 it does not. Somebody's missing something, I guess.
wachovia: I don't remember saying anything about presumptions or conditions, I've always just quoted odds and how they apply to a die roll. I read your post twice and actually started answering each of your assertions before coming to the conclusion I don't get how flipping a coin applies to a die roll (guess that puts me in the company of those folks who didn't get how the Monty Hall game show scenario applies to die rolls). Sorry. Maybe you've identified how superior you might be in the intelligence department (and, yes, I know that just threw a softball to those of you who have been questioning my intelligence).
Tautology? Wow - a true Verbal Advantage word. I had to look it up. From Wikipedia:
Tautology (rhetoric), repetition of meaning, using dissimilar words to say the same thing twice, especially where the additional words fail to provide additional clarity and meaning.
Tautology (logic), a technical notion in formal logic, universal unconditioned truth, always valid.
If you're applying either of those to me, I think you give me far too much credit.
Very entertaining clip, Turtle. Rmilter would love that with his entirely same column lineup.
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by MARCPELLETIER » Tue Sep 15, 2009 12:55 pm
ugrant,
why do you limit yourself to two rolls. What happens with the third roll?
What about if offensive cards had six columns. My understanding of your reasoning is that, after five straight misses, you would be 100% sure to hit the good column with the sixth card.
why do you limit yourself to two rolls. What happens with the third roll?
What about if offensive cards had six columns. My understanding of your reasoning is that, after five straight misses, you would be 100% sure to hit the good column with the sixth card.
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by crs76 » Tue Sep 15, 2009 1:35 pm
ugrant, what is you ideal set up for your first 3 batters.
Leadoff column 1, 2nd column 2, 3rd column 3???
Leadoff column 3, 2nd column 1, 3rd column 2???
or
Leadoff column 1, 2nd column 1, 3rd column 1???
No matter which one you choose your team will have a 1 in 216 (6x6x6) chance of leading off the game with the first 3 rolls on your hitter's "good" column.
Hope this helps...
Leadoff column 1, 2nd column 2, 3rd column 3???
Leadoff column 3, 2nd column 1, 3rd column 2???
or
Leadoff column 1, 2nd column 1, 3rd column 1???
No matter which one you choose your team will have a 1 in 216 (6x6x6) chance of leading off the game with the first 3 rolls on your hitter's "good" column.
Hope this helps...
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by ugrant » Tue Sep 15, 2009 2:01 pm
Marcus: my posts have only dealt with the odds of sequential columns occurring one roll after another (two rolls). If your understanding of what I have posted leads you to any conclusion of what would happen after five rolls, I would postulate you haven't actually read all of my posts.
I'll summarize. Every roll is an independent event. Each column of the six represented by the six sides of a die have a 1/6 chance of being rolled (every roll). No one disagrees with that and almost everyone who has posted here leaves it right there (and keeps trying to show me I'm wrong by taking the next step).
Which is: after a column has been rolled, the chances of that column being rolled again in succession are 1/6 (16.6%). The chances of it not being rolled are 5/6 (83.3%). The player who aligns his cards in his lineup with similar "good" columns in sequence has a 1/6 chance of having those "good" columns rolled in sequence (when the previous roll has, in fact, resulted in that "good" column being rolled - that's for wachovia, btw, who "presumed" I thought every previous roll to have been "good"). For that player, the odds of having "good" columns rolled sequentially will be 16.7%.
The player who alternates his lineup so that "good" columns do not replicate (as much as possible, it happens sometimes it can't be done based on the players cards in the lineup) is taking the odds that the 5/6 chance of the roll not duplicating occurs - the 83.3% chance. If that occurs, then there are five columns left. The alternating column player now has a 1/5 chance of having the "good" column rolled (20%).
The rub is when to apply the 83.3% to the math of the odds. Just about everyone here (remember, I didn't create this thread), says it always applies, that 83.3% X 20% = 16.7% and that's that, every roll (column) has a 16.7% chance of occurring (I have no issue with that when determining the odds of an individual column being rolled).
I say the 83.3% determines whether or not the roll duplicates the previous roll (column), and if it occurs that the roll does not duplicate the previous roll (column), then the 20% becomes stand alone odds (1 of 5 remaining columns).
An 83.3% chance of having a 20% chance of rolling a "good" column after a previous "good" column was rolled.
The next roll (being the third roll, which is what I think you're getting at) would be looked at with exactly the same odds based on only the roll prior (not the last two or three rolls).
I'll summarize. Every roll is an independent event. Each column of the six represented by the six sides of a die have a 1/6 chance of being rolled (every roll). No one disagrees with that and almost everyone who has posted here leaves it right there (and keeps trying to show me I'm wrong by taking the next step).
Which is: after a column has been rolled, the chances of that column being rolled again in succession are 1/6 (16.6%). The chances of it not being rolled are 5/6 (83.3%). The player who aligns his cards in his lineup with similar "good" columns in sequence has a 1/6 chance of having those "good" columns rolled in sequence (when the previous roll has, in fact, resulted in that "good" column being rolled - that's for wachovia, btw, who "presumed" I thought every previous roll to have been "good"). For that player, the odds of having "good" columns rolled sequentially will be 16.7%.
The player who alternates his lineup so that "good" columns do not replicate (as much as possible, it happens sometimes it can't be done based on the players cards in the lineup) is taking the odds that the 5/6 chance of the roll not duplicating occurs - the 83.3% chance. If that occurs, then there are five columns left. The alternating column player now has a 1/5 chance of having the "good" column rolled (20%).
The rub is when to apply the 83.3% to the math of the odds. Just about everyone here (remember, I didn't create this thread), says it always applies, that 83.3% X 20% = 16.7% and that's that, every roll (column) has a 16.7% chance of occurring (I have no issue with that when determining the odds of an individual column being rolled).
I say the 83.3% determines whether or not the roll duplicates the previous roll (column), and if it occurs that the roll does not duplicate the previous roll (column), then the 20% becomes stand alone odds (1 of 5 remaining columns).
An 83.3% chance of having a 20% chance of rolling a "good" column after a previous "good" column was rolled.
The next roll (being the third roll, which is what I think you're getting at) would be looked at with exactly the same odds based on only the roll prior (not the last two or three rolls).
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by MARCPELLETIER » Tue Sep 15, 2009 2:22 pm
[quote:5ec75dcec4]
The next roll (being the third roll, which is what I think you're getting at) would be looked at with exactly the same odds based on only the roll prior (not the last two or three rolls). [/quote:5ec75dcec4]
Well, I think your reasoning should force you to look at least at three consecutive rolls, not simply two rolls.
Surely, again, if I follow your logic, you have some better odds if you go with the following line-up:
(scenario A)
leadoff: column X
2nd hitter: column Y
3rd hitter: column Z
than if you go with:
(scenario B)
leadoff: column X
2nd hitter: column Y
3rd hitter: column X
With scenario A, you have a 67% chance of having a 25% odds to not replicate the two preceding results. With scenario B, you have a 100% chance of a 16.7% odds to replicate one of the two preceding results.
My understanding: you prefer scenario A to scenario B.
The next roll (being the third roll, which is what I think you're getting at) would be looked at with exactly the same odds based on only the roll prior (not the last two or three rolls). [/quote:5ec75dcec4]
Well, I think your reasoning should force you to look at least at three consecutive rolls, not simply two rolls.
Surely, again, if I follow your logic, you have some better odds if you go with the following line-up:
(scenario A)
leadoff: column X
2nd hitter: column Y
3rd hitter: column Z
than if you go with:
(scenario B)
leadoff: column X
2nd hitter: column Y
3rd hitter: column X
With scenario A, you have a 67% chance of having a 25% odds to not replicate the two preceding results. With scenario B, you have a 100% chance of a 16.7% odds to replicate one of the two preceding results.
My understanding: you prefer scenario A to scenario B.
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by RICHARDMILTER » Tue Sep 15, 2009 2:25 pm
Anyone want to go to Vegas this Winter? Maybe have a little Strato tournament out there? I was thinking early January for NFL Wild Card weekend.
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by keyzick » Tue Sep 15, 2009 2:58 pm
[quote:95d3378181="ugrant"]The rub is when to apply the 83.3% to the math of the odds. Just about everyone here (remember, I didn't create this thread), says it always applies, that 83.3% X 20% = 16.7% and that's that, every roll (column) has a 16.7% chance of occurring (I have no issue with that when determining the odds of an individual column being rolled).[/quote:95d3378181]
OK, sounds like all in agreement to this point.
Now, for the remainder of that post:
[quote:95d3378181="ugrant"]I say the 83.3% determines whether or not the roll duplicates the previous roll (column), and if it occurs that the roll does not duplicate the previous roll (column), then the 20% becomes stand alone odds (1 of 5 remaining columns).
An 83.3% chance of having a 20% chance of rolling a "good" column after a previous "good" column was rolled.[/quote:95d3378181]
[b:95d3378181][i:95d3378181]Your last sentence, is mathematically restating the same 16.7% again.[/i:95d3378181][/b:95d3378181]
Wachovia's post was the most thorough and accurately detailed statement of all...not sure why we took anything disparaging from his comments.
OK, sounds like all in agreement to this point.
Now, for the remainder of that post:
[quote:95d3378181="ugrant"]I say the 83.3% determines whether or not the roll duplicates the previous roll (column), and if it occurs that the roll does not duplicate the previous roll (column), then the 20% becomes stand alone odds (1 of 5 remaining columns).
An 83.3% chance of having a 20% chance of rolling a "good" column after a previous "good" column was rolled.[/quote:95d3378181]
[b:95d3378181][i:95d3378181]Your last sentence, is mathematically restating the same 16.7% again.[/i:95d3378181][/b:95d3378181]
Wachovia's post was the most thorough and accurately detailed statement of all...not sure why we took anything disparaging from his comments.
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by ugrant » Tue Sep 15, 2009 3:01 pm
Marcus - I really haven't thought about successive rolls beyond two and at the moment have neither the time or desire to. If you want to go down that road, be my guest.
I do appreciate your civility, whatever you might think of this subject.
I do appreciate your civility, whatever you might think of this subject.
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