• Log InLog In
  • Register
Liquid`
Team Liquid Liquipedia
EDT 23:08
CEST 05:08
KST 12:08
  • Home
  • Forum
  • Calendar
  • Streams
  • Liquipedia
  • Features
  • Store
  • EPT
  • TL+
  • StarCraft 2
  • Brood War
  • Smash
  • Heroes
  • Counter-Strike
  • Overwatch
  • Liquibet
  • Fantasy StarCraft
  • TLPD
  • StarCraft 2
  • Brood War
  • Blogs
Forum Sidebar
Events/Features
News
Featured News
[ASL20] Ro16 Preview Pt1: Ascent8Maestros of the Game: Week 1/Play-in Preview12[ASL20] Ro24 Preview Pt2: Take-Off7[ASL20] Ro24 Preview Pt1: Runway132v2 & SC: Evo Complete: Weekend Double Feature4
Community News
Weekly Cups (Sept 1-7): MaxPax rebounds & Clem saga continues9LiuLi Cup - September 2025 Tournaments2Weekly Cups (August 25-31): Clem's Last Straw?39Weekly Cups (Aug 18-24): herO dethrones MaxPax6Maestros of The Game—$20k event w/ live finals in Paris73
StarCraft 2
General
Weekly Cups (Sept 1-7): MaxPax rebounds & Clem saga continues Team Liquid Map Contest #21 - Presented by Monster Energy What happened to Singapore/Brazil servers? #1: Maru - Greatest Players of All Time Production Quality - Maestros of the Game Vs RSL 2
Tourneys
Maestros of The Game—$20k event w/ live finals in Paris Sparkling Tuna Cup - Weekly Open Tournament RSL: Revival, a new crowdfunded tournament series Chzzk MurlocKing SC1 vs SC2 Cup Sea Duckling Open (Global, Bronze-Diamond)
Strategy
Custom Maps
External Content
Mutation # 490 Masters of Midnight Mutation # 489 Bannable Offense Mutation # 488 What Goes Around Mutation # 487 Think Fast
Brood War
General
FlaSh on ACS Winners being in ASL ASL20 General Discussion BGH Auto Balance -> http://bghmmr.eu/ BSL Polish World Championship 2025 20-21 September [ASL20] Ro16 Preview Pt1: Ascent
Tourneys
[ASL20] Ro16 Group A [IPSL] ISPL Season 1 Winter Qualis and Info! [Megathread] Daily Proleagues Is there English video for group selection for ASL
Strategy
Simple Questions, Simple Answers Muta micro map competition Fighting Spirit mining rates [G] Mineral Boosting
Other Games
General Games
Stormgate/Frost Giant Megathread Nintendo Switch Thread General RTS Discussion Thread Path of Exile Warcraft III: The Frozen Throne
Dota 2
Official 'what is Dota anymore' discussion
League of Legends
Heroes of the Storm
Simple Questions, Simple Answers Heroes of the Storm 2.0
Hearthstone
Heroes of StarCraft mini-set
TL Mafia
TL Mafia Community Thread Vanilla Mini Mafia
Community
General
US Politics Mega-thread Things Aren’t Peaceful in Palestine The Games Industry And ATVI UK Politics Mega-thread Russo-Ukrainian War Thread
Fan Clubs
The Happy Fan Club!
Media & Entertainment
[Manga] One Piece Anime Discussion Thread Movie Discussion! [\m/] Heavy Metal Thread
Sports
2024 - 2026 Football Thread Formula 1 Discussion MLB/Baseball 2023 TeamLiquid Health and Fitness Initiative For 2023
World Cup 2022
Tech Support
Computer Build, Upgrade & Buying Resource Thread High temperatures on bridge(s)
TL Community
The Automated Ban List TeamLiquid Team Shirt On Sale
Blogs
Collective Intelligence: Tea…
TrAiDoS
A very expensive lesson on ma…
Garnet
hello world
radishsoup
Lemme tell you a thing o…
JoinTheRain
RTS Design in Hypercoven
a11
Evil Gacha Games and the…
ffswowsucks
INDEPENDIENTE LA CTM
XenOsky
Customize Sidebar...

Website Feedback

Closed Threads



Active: 1221 users

The Law of Averages - Page 2

Blogs > obesechicken13
Post a Reply
Prev 1 2 All
MoonBear
Profile Blog Joined November 2010
Straight outta Johto18973 Posts
May 20 2011 22:57 GMT
#21
The statistical fallacies in this blog makes me sad. I understand how the message is supposed to work, but the execution was incredibly dubious.
ModeratorA dream. Do you have one that has cursed you like that? Or maybe... a wish?
Lemonwalrus
Profile Blog Joined August 2006
United States5465 Posts
May 20 2011 22:59 GMT
#22
Remake the blog, the more you try, the more likely you are to get it right sooner or later.

I learned that from a blog once.
EsX_Raptor
Profile Blog Joined February 2008
United States2801 Posts
May 21 2011 01:24 GMT
#23
Do you understand that there is a probability that you will never roll a 6 after an infinite amount of casts?
obesechicken13
Profile Blog Joined July 2008
United States10467 Posts
May 21 2011 02:15 GMT
#24
On May 21 2011 07:59 Lemonwalrus wrote:
Remake the blog, the more you try, the more likely you are to get it right sooner or later.

I learned that from a blog once.


Eh, you guys figured out what I was saying eventually.

On May 21 2011 10:24 EsX_Raptor wrote:
Do you understand that there is a probability that you will never roll a 6 after an infinite amount of casts?


No there isn't . pfft infinity.
I think in our modern age technology has evolved to become more addictive. The things that don't give us pleasure aren't used as much. Work was never meant to be fun, but doing it makes us happier in the long run.
Sleight
Profile Blog Joined May 2009
2471 Posts
Last Edited: 2011-05-21 02:25:36
May 21 2011 02:21 GMT
#25
Guys, you really don't understand probability theory.

Two dice, any chance of getting one single 6 is the same as rolling one die twice to get a six.

6 * 6 = 36 Outcomes
Winning outcomes = [6 X] and [X 6] with [6 6] only counting once, obviously, so 11 wins.
11 / 36 = .306

You guys need to do better math. Rolling more dice DOES increase the probability of a success. Taking more chances increases the likelihood of a success as a strategy.

EDIT for clarification: No single roll EVER has a higher likelihood than 1/6, but you can only examine probability on the large scale. We are looking at combinations of chances, which mean that if you roll 1000 dice, your likelihood of getting a 6 is basically 1, but that says nothing of whether its the first roll or the last roll. Probability theory ONLY applies to SETS.
One Love
Empyrean
Profile Blog Joined September 2004
16992 Posts
Last Edited: 2011-05-21 03:09:07
May 21 2011 03:03 GMT
#26
On May 21 2011 10:24 EsX_Raptor wrote:
Do you understand that there is a probability that you will never roll a 6 after an infinite amount of casts?


The probability is zero, because you will almost surely roll a 6 in an infinite amount of casts.

The amount of statistical ignorance in the general population is staggering

EDIT: Before someone replies, just because an event occurs with probability zero doesn't mean it can't occur. And I'm using the term "almost surely" in its technical meaning, in case there were any doubts.
Moderator
MoonBear
Profile Blog Joined November 2010
Straight outta Johto18973 Posts
May 21 2011 03:15 GMT
#27
On May 21 2011 11:21 Sleight wrote:
Guys, you really don't understand probability theory.

Two dice, any chance of getting one single 6 is the same as rolling one die twice to get a six.

6 * 6 = 36 Outcomes
Winning outcomes = [6 X] and [X 6] with [6 6] only counting once, obviously, so 11 wins.
11 / 36 = .306

You guys need to do better math. Rolling more dice DOES increase the probability of a success. Taking more chances increases the likelihood of a success as a strategy.

EDIT for clarification: No single roll EVER has a higher likelihood than 1/6, but you can only examine probability on the large scale. We are looking at combinations of chances, which mean that if you roll 1000 dice, your likelihood of getting a 6 is basically 1, but that says nothing of whether its the first roll or the last roll. Probability theory ONLY applies to SETS.

I'm not sure what point you are trying to make with your first example of rolling two dice concurrently. But if you are saying that the chance getting a 6 when rolling two dice at the same time is different from the probability of obtaining a 6 when rolling a single die is different then the result is a given. In that instance you are comparing a joint probability distribution to a single variable probability distribution so the results will be skewed as a result.

You are misunderstanding the point that Infinity was making. There is nothing wrong with observing probability on a small scale. Statisticians do this all the time.

The "Law of Large Numbers" (I do detest the name but no matter) is a statement that a very large number of observations will tend to reflect the underlying probability of the random variable in question. On this instance, I believe we are all in agreement. If you flip a fair coin an infinite number of times, your observations will tend to a 50:50 split.

However, you are confusing this with the applications and caveats that come with independence and random sampling. The OP is not making a statement about deducing conclusion from an extremely large sample space. Instead, it is a case of repeated trials and commenting on the results of each observation. In this case, it is a conditional probability function.

For example, one example was continuing to hit on women at a party until you achieve success. It would be incorrect to state that we are making observations on a large scale. Instead, you are taking repeated samples from a state space and then commenting on each observation one at a time. The fallacy in his article was "Given that I have made advances on a woman and failed, I expect to have a success because the law of averages should even things out eventually." This is incorrect because independence between events means that the conditional probability of success given failure is still the same as the probability of success in a single event.

I would recommend you read the link that Infinity posted and not make claims such as "You don't understand probability theory".
ModeratorA dream. Do you have one that has cursed you like that? Or maybe... a wish?
Sleight
Profile Blog Joined May 2009
2471 Posts
May 21 2011 03:25 GMT
#28
I pointed out exactly how the Gambler's fallacy doesn't account for reality, friend. I also said probability only works on SETS as my last statement. Two instances is a set, even if its largely useless. Two hundred instances and we begin to find relevant information to inform a decision, typically.

Notice how I explicitly say your single instance probably NEVER CHANGES?

Did you read my post before saying the same thing in a different way and trying to sound contradictory? I said nothing about the OP's correctness in regards to his Gambler's Fallacy. I spoke only to the event that increased trials increases chance of success. He is right, statistically you will eventually hit it home, but he is wrong in that it is some magical self-correction. It's just probability, nothing more, nothing less.

Like I said and you said and other informed people have said.
One Love
MoonBear
Profile Blog Joined November 2010
Straight outta Johto18973 Posts
May 21 2011 03:39 GMT
#29
On May 21 2011 12:25 Sleight wrote:
I pointed out exactly how the Gambler's fallacy doesn't account for reality, friend. I also said probability only works on SETS as my last statement. Two instances is a set, even if its largely useless. Two hundred instances and we begin to find relevant information to inform a decision, typically.

Notice how I explicitly say your single instance probably NEVER CHANGES?

Did you read my post before saying the same thing in a different way and trying to sound contradictory? I said nothing about the OP's correctness in regards to his Gambler's Fallacy. I spoke only to the event that increased trials increases chance of success. He is right, statistically you will eventually hit it home, but he is wrong in that it is some magical self-correction. It's just probability, nothing more, nothing less.

Like I said and you said and other informed people have said.

There are two disagreements with your argument I have.

The first is "Rolling more dice DOES increase the probability of a success." If we consider rolling die to be a Bernoulli or Binomial process, then each roll has the same probability. The probability of success on each roll hasn't changed. Increased trials does not "increase the chance of success" as you say. It simply reduces the effect of randomness and chance from your samples in the long run. What repeated samples means is that the distribution of your sample results will converge to the underlying distribution.

The second disagreement is that "you can only examine probability on the large scale". You can model any event regardless of how large your sample size is.
ModeratorA dream. Do you have one that has cursed you like that? Or maybe... a wish?
Sleight
Profile Blog Joined May 2009
2471 Posts
May 21 2011 04:03 GMT
#30
You have quit reading what I said, clearly.

Again, for the record. Increasing trials DOES NOT increase the probability of ANY SINGLE INSTANCE.

Got it? I've said that each and every post. Rolling a die 10k times does not increase the chance of getting a 6 on any given trial.

You are wrong to say that the probability of getting a 6 rolling a dice once is the same as rolling a dice 10k times. Dead wrong. Which is what I have said over and over. You are COMPLETELY WRONG if defining this as a Bernoulli process formally changes ANYTHING from what I've said.

Let's define our process as such. If a die rolls a non-6, we score it a 0. If the die is a 6 it is a 1, p = 1/6. Let's run a trial on that for one time.

We will say P(1) = (1 choose 1)*.16^1*.84*0 = .16. The chance of getting a 6 in 1 roll.

We will ask what are the chances of getting at least 1 6 in 2 trials.
P(1)=(2 choose 1)*.16^1*.84^1 = .268.

Does this account for all outcomes? NO! We still need to consider 2 successes.
P(2)=(2 choose2)*.16^2*.84^1 = .022

P of at least 1 success = .268+.022 = .29 (i rounded .166 to .16, so its marginally lower).

You are unequivocally wrong that changing the model somehow stops more trials from increasing chance of success.
One Love
obesechicken13
Profile Blog Joined July 2008
United States10467 Posts
Last Edited: 2011-05-21 04:37:51
May 21 2011 04:30 GMT
#31
On May 21 2011 12:15 MoonBear wrote:
Show nested quote +
On May 21 2011 11:21 Sleight wrote:
Guys, you really don't understand probability theory.

Two dice, any chance of getting one single 6 is the same as rolling one die twice to get a six.

6 * 6 = 36 Outcomes
Winning outcomes = [6 X] and [X 6] with [6 6] only counting once, obviously, so 11 wins.
11 / 36 = .306

You guys need to do better math. Rolling more dice DOES increase the probability of a success. Taking more chances increases the likelihood of a success as a strategy.

EDIT for clarification: No single roll EVER has a higher likelihood than 1/6, but you can only examine probability on the large scale. We are looking at combinations of chances, which mean that if you roll 1000 dice, your likelihood of getting a 6 is basically 1, but that says nothing of whether its the first roll or the last roll. Probability theory ONLY applies to SETS.

I'm not sure what point you are trying to make with your first example of rolling two dice concurrently. But if you are saying that the chance getting a 6 when rolling two dice at the same time is different from the probability of obtaining a 6 when rolling a single die is different then the result is a given. In that instance you are comparing a joint probability distribution to a single variable probability distribution so the results will be skewed as a result.

You are misunderstanding the point that Infinity was making. There is nothing wrong with observing probability on a small scale. Statisticians do this all the time.

The "Law of Large Numbers" (I do detest the name but no matter) is a statement that a very large number of observations will tend to reflect the underlying probability of the random variable in question. On this instance, I believe we are all in agreement. If you flip a fair coin an infinite number of times, your observations will tend to a 50:50 split.

However, you are confusing this with the applications and caveats that come with independence and random sampling. The OP is not making a statement about deducing conclusion from an extremely large sample space. Instead, it is a case of repeated trials and commenting on the results of each observation. In this case, it is a conditional probability function.

For example, one example was continuing to hit on women at a party until you achieve success. It would be incorrect to state that we are making observations on a large scale. Instead, you are taking repeated samples from a state space and then commenting on each observation one at a time.


Hmmm... ok... ok


The fallacy in his article was "Given that I have made advances on a woman and failed, I expect to have a success because the law of averages should even things out eventually." This is incorrect because independence between events means that the conditional probability of success given failure is still the same as the probability of success in a single event.

I would recommend you read the link that Infinity posted and not make claims such as "You don't understand probability theory".

Hey I never said that!
+ Show Spoiler +
;_; Even if it is true


And I quote
The law of averages is actually not a law at all. It's a truism that has no truth stating that when you've flipped a coin and gotten heads three times in a row, well then tails is more likely to turn up on your next flip. I just think that "The Law of Large Numbers" is a horrible title.


My intention wasn't to say that I would have a greater chance of success on my next attempt, I was saying that if I got turned down 20 times, and then approached 50 more women, I would have a higher chance of success with at least one of those 50 women than if I felt discouraged and only approached two women. I'm trying to say don't give up. I understand "independece between events" or whatever
+ Show Spoiler +
[image loading]
Hmpf

I think in our modern age technology has evolved to become more addictive. The things that don't give us pleasure aren't used as much. Work was never meant to be fun, but doing it makes us happier in the long run.
Cambium
Profile Blog Joined June 2004
United States16368 Posts
May 21 2011 04:41 GMT
#32
I was just going to type up something, then I saw two pages of comments

<3 TL
When you want something, all the universe conspires in helping you to achieve it.
infinity21 *
Profile Blog Joined October 2006
Canada6683 Posts
May 21 2011 05:44 GMT
#33
You guys are literally arguing over the difference between 'chances of success' and 'chances of a success' lol
Official Entusman #21
Empyrean
Profile Blog Joined September 2004
16992 Posts
May 21 2011 06:37 GMT
#34
On May 21 2011 14:44 infinity21 wrote:
You guys are literally arguing over the difference between 'chances of success' and 'chances of a success' lol


It's not worth trying.
Moderator
xxpack09
Profile Blog Joined September 2010
United States2160 Posts
May 21 2011 06:55 GMT
#35
You guys are missing the whole point of the blog. To use the dice analogy, the point is if you roll the die 100 times you're more likely to get a 6 in AT LEAST one of those 100 rolls than if you roll the die once. Which he applies to things like hooking up with girls, getting a job, etc.
Empyrean
Profile Blog Joined September 2004
16992 Posts
May 21 2011 09:56 GMT
#36
Right. But that's not what the law of averages is.
Moderator
Mactator
Profile Joined March 2011
109 Posts
Last Edited: 2011-05-21 10:35:07
May 21 2011 10:22 GMT
#37
On May 21 2011 13:30 obesechicken13 wrote:
Show nested quote +
On May 21 2011 12:15 MoonBear wrote:
On May 21 2011 11:21 Sleight wrote:
Guys, you really don't understand probability theory.

Two dice, any chance of getting one single 6 is the same as rolling one die twice to get a six.

6 * 6 = 36 Outcomes
Winning outcomes = [6 X] and [X 6] with [6 6] only counting once, obviously, so 11 wins.
11 / 36 = .306

You guys need to do better math. Rolling more dice DOES increase the probability of a success. Taking more chances increases the likelihood of a success as a strategy.

EDIT for clarification: No single roll EVER has a higher likelihood than 1/6, but you can only examine probability on the large scale. We are looking at combinations of chances, which mean that if you roll 1000 dice, your likelihood of getting a 6 is basically 1, but that says nothing of whether its the first roll or the last roll. Probability theory ONLY applies to SETS.

I'm not sure what point you are trying to make with your first example of rolling two dice concurrently. But if you are saying that the chance getting a 6 when rolling two dice at the same time is different from the probability of obtaining a 6 when rolling a single die is different then the result is a given. In that instance you are comparing a joint probability distribution to a single variable probability distribution so the results will be skewed as a result.

You are misunderstanding the point that Infinity was making. There is nothing wrong with observing probability on a small scale. Statisticians do this all the time.

The "Law of Large Numbers" (I do detest the name but no matter) is a statement that a very large number of observations will tend to reflect the underlying probability of the random variable in question. On this instance, I believe we are all in agreement. If you flip a fair coin an infinite number of times, your observations will tend to a 50:50 split.

However, you are confusing this with the applications and caveats that come with independence and random sampling. The OP is not making a statement about deducing conclusion from an extremely large sample space. Instead, it is a case of repeated trials and commenting on the results of each observation. In this case, it is a conditional probability function.

For example, one example was continuing to hit on women at a party until you achieve success. It would be incorrect to state that we are making observations on a large scale. Instead, you are taking repeated samples from a state space and then commenting on each observation one at a time.


Hmmm... ok... ok

Show nested quote +

The fallacy in his article was "Given that I have made advances on a woman and failed, I expect to have a success because the law of averages should even things out eventually." This is incorrect because independence between events means that the conditional probability of success given failure is still the same as the probability of success in a single event.

I would recommend you read the link that Infinity posted and not make claims such as "You don't understand probability theory".

Hey I never said that!
+ Show Spoiler +
;_; Even if it is true


And I quote
Show nested quote +
The law of averages is actually not a law at all. It's a truism that has no truth stating that when you've flipped a coin and gotten heads three times in a row, well then tails is more likely to turn up on your next flip. I just think that "The Law of Large Numbers" is a horrible title.


My intention wasn't to say that I would have a greater chance of success on my next attempt, I was saying that if I got turned down 20 times, and then approached 50 more women, I would have a higher chance of success with at least one of those 50 women than if I felt discouraged and only approached two women. I'm trying to say don't give up. I understand "independece between events" or whatever
+ Show Spoiler +
[image loading]
Hmpf



Oh I wish you were right and that success in life was proportional to the number of times you try. The problem though is that if you fuck up badly then the probability to succeed next time you try will decrease. Or the other way around. Real life events usually aren't independent and you can fail if you behave stupidly :-).

EDIT: One may say that this is true for sc2 as well. Trying a lot is a good thing but if you don't analyze your games, optimize your build orders and create new ones well.. then there is a chance that you aren't improving or perhaps even get bad habits.
Prev 1 2 All
Please log in or register to reply.
Live Events Refresh
Next event in 6h 52m
[ Submit Event ]
Live Streams
Refresh
StarCraft 2
Nina 208
Nathanias 92
ProTech67
CosmosSc2 52
StarCraft: Brood War
Shuttle 402
Hyuk 321
sSak 32
Bale 14
Icarus 10
Dota 2
monkeys_forever1130
NeuroSwarm102
LuMiX0
League of Legends
JimRising 835
Counter-Strike
fl0m1527
taco 230
semphis_33
Super Smash Bros
C9.Mang0437
Other Games
summit1g5887
shahzam1104
Day[9].tv348
Maynarde143
WinterStarcraft47
RuFF_SC223
Organizations
Other Games
gamesdonequick1992
StarCraft 2
CranKy Ducklings110
Blizzard YouTube
StarCraft: Brood War
BSLTrovo
sctven
[ Show 15 non-featured ]
StarCraft 2
• AfreecaTV YouTube
• intothetv
• Kozan
• IndyKCrew
• LaughNgamezSOOP
• Migwel
• sooper7s
StarCraft: Brood War
• Mapu5
• BSLYoutube
• STPLYoutube
• ZZZeroYoutube
Dota 2
• Ler33
League of Legends
• Lourlo287
Other Games
• Scarra1106
• Day9tv348
Upcoming Events
Afreeca Starleague
6h 52m
BeSt vs Alone
Queen vs Bisu
Kung Fu Cup
8h 52m
PiGosaur Monday
20h 52m
Kung Fu Cup
1d 8h
OSC
1d 12h
OSC
1d 20h
RSL Revival
2 days
Cure vs SHIN
Reynor vs Zoun
Kung Fu Cup
2 days
The PondCast
2 days
RSL Revival
3 days
Classic vs TriGGeR
ByuN vs Maru
[ Show More ]
Online Event
3 days
Kung Fu Cup
3 days
BSL Team Wars
3 days
Team Bonyth vs Team Dewalt
BSL Team Wars
3 days
RSL Revival
4 days
Maestros of the Game
4 days
ShoWTimE vs Classic
Clem vs herO
Serral vs Bunny
Reynor vs Zoun
Cosmonarchy
4 days
Bonyth vs Dewalt
[BSL 2025] Weekly
4 days
RSL Revival
5 days
Maestros of the Game
5 days
BSL Team Wars
5 days
Afreeca Starleague
6 days
Snow vs Sharp
Jaedong vs Mini
Liquipedia Results

Completed

Copa Latinoamericana 4
SEL Season 2 Championship
HCC Europe

Ongoing

BSL 20 Team Wars
KCM Race Survival 2025 Season 3
BSL 21: BSL Points
ASL Season 20
CSL 2025 AUTUMN (S18)
LASL Season 20
RSL Revival: Season 2
Maestros of the Game
Chzzk MurlocKing SC1 vs SC2 Cup #2
BLAST Open Fall 2025
BLAST Open Fall Qual
Esports World Cup 2025
BLAST Bounty Fall 2025
BLAST Bounty Fall Qual
IEM Cologne 2025
FISSURE Playground #1

Upcoming

2025 Chongqing Offline CUP
BSL Polish World Championship 2025
BSL Season 21
BSL 21 Team A
EC S1
BLAST Rivals Fall 2025
IEM Chengdu 2025
PGL Masters Bucharest 2025
MESA Nomadic Masters Fall
Thunderpick World Champ.
CS Asia Championships 2025
ESL Pro League S22
StarSeries Fall 2025
FISSURE Playground #2
TLPD

1. ByuN
2. TY
3. Dark
4. Solar
5. Stats
6. Nerchio
7. sOs
8. soO
9. INnoVation
10. Elazer
1. Rain
2. Flash
3. EffOrt
4. Last
5. Bisu
6. Soulkey
7. Mini
8. Sharp
Sidebar Settings...

Advertising | Privacy Policy | Terms Of Use | Contact Us

Original banner artwork: Jim Warren
The contents of this webpage are copyright © 2025 TLnet. All Rights Reserved.