Sunday, September 16, 2007

Arithmetic Emprical not Truth

The problem in arithmetic is people use applications to physical phenomena without the necessary experience to understand if it even applies. Examples of arithmetic applications can be absurd. I like Grey Goose Vodka and if a quart of alcohol is mixed with a quart of water it will yield ~1.8 quarts of vodka, which is true of most mixtures of alcoholic liquids. I play basketball and if one night I hit 2 out of 3 shots and the next night I make 3 out of 4 shots and you compute my average over both games by adding the two fractions in the usual method of adding fractions, which is find the common denominator:
2/3 +3/4 = 8/12+9/12=17/12
So, I made 17 shots in 12 tries? The method of adding fractions does not give my shooting average over separate games.

A new way of adding fractions is add the numerators and add the denominators:
2/3 +3/4=5/7
Experience tells us this arithmetic method is correct.

Under normal rules 2/3=4/6 but not under this new method of adding fractions:
4/6+3/4=7/10, and 7/10 does NOT equal 5/7.

In normal arithmetic, fractions behave as integers. So consider the numerators 2 and 4. The numbers can be fractions: 2/1 and 4/1.

Now use our new arithmetic:
2/1+4/1=6/2 rather than the 6/1 obtained under normal rules.

This has a major significance since it concludes that there is no truth in mathematics but simply a tool developed by man. Only experience can show where ordinary arithmetic applies to any given physical phenomena. The theorems of mathematics are not truths but man's tool to model aspects about the physical world. As far as the study of the physical world is concerned, mathematics merely offers theories. Mathematics is based on man's intuition whereas axioms of logic or truth is an art where man is allowed to paint his better wrong answer with confidence.

Fractions and Odds

To convert odds to fractions: a to b is b/ (a+b)
To convert fractions to odds: a/b is (b-a) to a

Odds are a ratio of non occurrences to occurrences (losses to wins).
Ratios reduce to lower terms [i.e. (9 to 6) = (3 to 2)].

Think of things in terms of fractions or percentages not odds.
3 to 1 shot: 1/ (1+3) = 1/4: 0.25 = 25%
hence 3 to 2 shot = 40%
4 to 1 shot: 1/ (1+4) = 1/5: 0.20 = 20%

Odds need to be converted to fractions to do any arithmetic. Change odds to percentages, add the percentages, and revert that to odds. (i.e. to parlay 3 to shot and 4 to 1 shot):
3 to 2: 2/5
4 to 1: 1/5
1/5 x 2/5 = 2/25, which reverts to 23 to 2 shot.

Percentage advantage is not the same as percentage disadvantage. If you know either then you can tell if your expectation is positive or negative. Your average profit per game is your expectations. With 9 to 1 odds, the house has a 1% advantage, for example each $110 bet makes them $1, yet the player disadvantage is to lose $1 for each $11 bet, which is 9.1% when he bets $1 per roll for eleven rolls where he wins once ($9) and they win $10.

Saturday, September 8, 2007

Mutually Exclusive vs Independent Events

Mutually exclusive or independent events must be considered before calculating the probability of two or more events occurring.

Two events "I am here" or "I am not here" are mutually exclusive. If ‘event A’ occurs, then ‘event B’ cannot, or vice-versa. To calculate the probabilities for exclusive events ADD the probabilities.

Two events "I am here" and "You are here" are independent. If ‘event A’, then if ‘event B’. To calculate the probabilities for independent events MULTIPLY the probabilities of two unrelated events.

The throw of a single die, outcomes are equally probable. Each throw is a mutually exclusive event. That is, if a die rolls A then it cannot roll B. The probability of rolling A or B with the first die is the sum of the equal probabilities, 1/6 + 1/6 = 1/3.
Independent events are multiplied. The probability of throwing two dice resulting in A and A is the probability of each roll times itself: (1/6) * (1/6) = 1/36th or 2.77%.

The probability of rolling A and B in either order with one dice is 1/3 x 1/6 = 1/18 (2/36). Add or multiply probabilities based on whether mutually exclusive or independent.

The different numbers obtained by the throw of two dice offer a good introduction to the ideas of probability. Possible outcomes for the total of two dice are not equal. Seven is a possible outcome with every throw of two dice. Two and 12 rolls only 1/36 (A and A) or (B and B). The "odds" of three (A or B) are two times greater 2/36. Rolling a 7 is six times greater 6/36, which raises the idea of distinguishable states. With two dice, the first dice throw does not matter as there remains a chance of rolling 7. If you throw one dice and it rolls a 3 then the next dice roll must be a 4 or vise-versa. Outcomes are exclusive if a result directly affects the other. The order does not matter (i.e. rolling a 3 is twice as likely as rolling a 2 since there are two distinguishable ways to get a 3). The probability of rolling a 7 with two dice is 1/6, rolled 6 of a total of 36 possible outcomes.