
Advanced Statistics Advanced Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 19th, 2010, 07:07 PM  #1 
Newbie Joined: Mar 2008 Posts: 15 Thanks: 0  Coin Flipping: Permutation vs. Combination
I appologize in advance. It was a struggle for me to write this question. I've been thinking about Gambler's Fallacy. As I understand things, every time you flip a coin or roll a dice, the odds reset. The coin or dice have no memory of the past. So if you flip a coin twice the odds aren't 1/4 to get two heads in a row, they are 50/50. The coin remains independent. If you flip a fair coin twice, on the first toss you have "2" possible Permutations? H T On the second toss "4" possible permutations HH HT TH TT The first toss has "2" possible results. A 50% chance of Heads. The second toss has "4" possible results. Still with a 50% chance of Heads. By the second toss, each Permutation has the same chances. HH 1/4 HT 1/4 TH 1/4 TT 1/4 The second throw has 2/4 Heads and 2/4 Tails. So the chances are 50% to get heads. If you increase the number of throws, the odds are the same for each Permutation. Out of 5 flips TTTTT is just as likely as THTHT. Although I'm not exactly sure these conclusions are correct, but I've been thinking about Combination versus Permutation. When you roll one six sided dice the result is independent. When you roll two sided dice, each result is independent. A dependency can be created by adding both together to get a range of 212. 
February 19th, 2010, 08:42 PM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Coin Flipping: Permutation vs. Combination Quote:
The gambler's fallacy is thinking that, if you've already flipped (say) HHH, you're more likely to get T than H because it's 'due'. Of course this is false: as you say, the coin has no memory.  
February 19th, 2010, 10:11 PM  #3 
Newbie Joined: Mar 2008 Posts: 15 Thanks: 0  Re: Coin Flipping: Permutation vs. Combination
Ok thanks for the answer. I think what throws me off is "the probability of any sequence is the same" or "every outcome observed will always have been equally as likely as the other outcomes that were not observed for that particular trial, given a fair coin. Therefore, just as Bayes' theorem shows, the result of each trial comes down to the base probability of the fair coin: 1/2." If the question is asked, "what are the odds the coin comes up heads on the second toss". The answer is 2/4. This would be looking for the idependence. When it is asked what are the odds of two TT by the second toss, they would be creating Dependence on the first throw. A coin toss is an Independent variable. The first and second. How do you "test" or demonstrate Independence? 
February 20th, 2010, 02:20 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Coin Flipping: Permutation vs. Combination Quote:
 
February 21st, 2010, 08:23 AM  #5 
Newbie Joined: Mar 2008 Posts: 15 Thanks: 0  Re: Coin Flipping: Permutation vs. Combination
Let me start over. Instead of labeling a coin Heads and Tails, lets call it 1 and 2; a two sided dice. If I flip it one time there are two possible outcomes: 1 and 2. If I flip it twice there are four possible outcomes: 11 12 21 22 If I combine the numbers I get a range from 24. I fully grasp that the odds of getting a total of "2" are 1/4 I fully grasp that the odds of getting a total of "4" are 2/4 I fully grasp that the odds of getting a total of "3" are 1/4 A bell curve is formed in the number distribution. Just like when 2d6 are rolled. The confusion stems from independence. It is declared a result of "1" is "pass" and a result of "2" is fail. Flip the coin once and you have a 50% to pass. Flip the coin three times and you have a 50% chance to pass on each attempt because the coin is Independent. You don't have a 1/8 chance of passing on the third try. There is "no memory" of past results. Another way to put it is playing a game like D&D. A player has a 50% skill to hit with a sword. First round the player has a 50% skill chance to hit. Lets say he rolls a hit. Second round the player still has a 50% skill chance. Since the dice have no memory, the chance is still the same as the first round. How is this expressed? Each time the coin (or dice) are used the odds are reset. Fundamental Counting Principle? Another example roulette wheel. Player 1 bets on black one time. Player 2 bets on black ten times. It doesn't matter because each spin of the wheel resets the odds. There is independence. Every permutation is just as likely. 
February 21st, 2010, 10:40 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Coin Flipping: Permutation vs. Combination Quote:
Quote:
Quote:
Quote:
 
February 21st, 2010, 12:36 PM  #7 
Newbie Joined: Mar 2008 Posts: 15 Thanks: 0  Re: Coin Flipping: Permutation vs. Combination
Ok. The RPG system has healing rolls of d100%. Depending on the severity your "Healing Rate" will be higher or lower. Like a 80% chance on a minor wound. If the d100% dice land on a "0" or "5" it is considered a critical of some sort. Example: 60% skill and you roll a "35", so that is a Critcal Success 60% skill and you roll a "41", so that is a Marginal Success 60% skill and you roll a "83", so that is a Marginal Failure 60% skill and you roll a "70", so that is a Critical Failure One method for the "healing process" is to roll daily. If you roll Critical Success the wound heals "2" points. If you roll a Marginal Success the wound heals "1" point. If you roll a Marginal Failure the wound doesn't heal any points. If you roll a Critical Failure the wound becomes Infected. The second method for the "healing process" is to roll for "5" day chunks. If you roll Critical Success the wound heals "10" points. If you roll a Marginal Success the wound heals "5" point. If you roll a Marginal Failure the wound doesn't heal any points. If you roll a Critical Failure the wound becomes Infected. If a character has a "20" point wound, does the "5 day chunk" method make it less likely to become infected? (rolling a critical failure) My observation is that both methods have the same chance of infection since each "trial" or "healing test" is independent of the previous roll. 
February 21st, 2010, 01:27 PM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Coin Flipping: Permutation vs. Combination Quote:
 
February 26th, 2010, 11:40 AM  #9  
Newbie Joined: Mar 2008 Posts: 15 Thanks: 0  Re: Coin Flipping: Permutation vs. Combination
I came across this. Quote:
The way I'm understanding things... If you throw 2d6 (one red the other green) and read them individually that would be an ordered set. If you throw 2d6 (one read the other green) and read them together that would be an unordered set. Isn't rolling 1d100 20 times and testing a skill an ordered set?  
February 26th, 2010, 07:00 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Coin Flipping: Permutation vs. Combination Quote:
1,1,3 is the same combination as 1,3,1, but they're different as permutations 1,1,3 has the same sum as 1,2,2, but they're different as combinations Quote:
If you're rolling with one of the two methods you described, stopping when * you recover all hit points, or * you roll a critical failure there is a particular distribution in terms of number of days required or eventual outcome.  

Tags 
coin, combination, flipping, permutation 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
A game with flipping a coin  SuperNova  Advanced Statistics  10  July 23rd, 2013 02:57 AM 
flipping a coin  aaronmath  Algebra  4  August 27th, 2012 09:46 AM 
Coin flipping game, approximation of one milion flips  mr_kaktus  Advanced Statistics  3  March 21st, 2011 06:10 AM 
Probability question[ coin flipping ]  tnutty  Probability and Statistics  3  January 27th, 2011 04:32 PM 
Interesting coin flipping!  johnny  Physics  4  August 6th, 2009 12:33 PM 