This is the definitive and the ultimate probability, gambling and

statistical software.

The program boasts several important formulae in theory of probability

and statistics:

1) The Fundamental Formula of Gambling (FFG: N from p and DC)

2) Degree of Certainty (DC from p and N)

3) Probability of FFG median (p from DC and N)

4) The Binomial Distribution Formula (BDF: EXACTLY M successes in N

trials)

5) The Probability of AT LEAST M successes in N trials

6) The Probability of AT MOST M successes in N trials

7) The Probability to WIN AT LEAST 'K of M in P from N' at Lotto &

PowerBall

8) The Binomial Standard Deviation (BSD)

9) Normal Probability Rule (more precise than Gauss curve)

10) Calculate Lotto Odds, For '0 of k' to 'm of k'

11) Hypergeometric Distribution Applied to Lotto Odds

12) Miscellanea: Sums of numbers, permutations, arrangements, shuffle numbers.

I. The Fundamental Formula of Gambling (FFG: N from p and DC)

This function applies the Fundamental Formula of Gambling (FFG). It

calculates the number of trials N necessary for an event of

probability p to appear with the degree of certainty DC.

For example, how many coin tosses are necessary to get at least one

'heads' (p = 1/2) with a degree of certainty equal to 99%? Answer: 7

tosses.

II. The Degree of Certainty (DC from p and N)

This function calculates the degree of certainty DC necessary for the

event of probability p to occur within N trials.

For example, what is the degree of certainty to get at least one

'heads' (p = 1/2) within 10 tosses? Answer: 99.902%.

III. The Probability of FFG Median (p from DC and N)

This function calculates the probability p when DC and N are known.

There are situations when you have the statistical median of a series

N; therefore DC=50%; but you don't know the probability of the

parameter p. The program calculates the probability p leading to a

degree of certainty DC and a number of trials N.

For example, the winning reports created by LotWon software show a

series of filters and their medians. If not calculated, you can use an

editor such as QEdit and do a column blocking, then sort the column

(filter) in descending order. The median represents the middle point

of the sorted column. The median also represents the number of trials

for a degree of certainty equal to 50%. I do not describe every filter

in my software, so nobody can tell the probability of every filter.

But you can determine it using this function of FORMULA.EXE. Other

filters are described and thus their probabilities can be calculated

in advance. They will prove the validity of the fundamental formula of

gambling (FFG). For example, the probability of '3 of 6' in a 6/49

lotto is 1 in 57. FFG calculates the median for this situation

(DC=50%) as 39. Take a real draw history, such as UK 6/49 lotto. Do

the winning report for 500 past draws. Sort in descending order the

filter "Threes" (or "3 #s") for layer 1. The median is 37 or closely

around 39. Reciprocally, when you see a median equal to 37, you can

determine the probability of the parameter as 1 in 54 (very close to

the real case of 1 in 57).

IV. The Binomial Distribution Formula (BDF)

The function calculates the probability BDF of exactly M successes in

N trials for an event of probability p.

For example, we want to determine the probability of getting exactly 5

"heads' in 10 tosses. We tossed the coin 7 times and recorded 5

"heads". We toss the coin for the 8th time and get another "heads"

(the 6th). We must stop the tossing; the experiment failed. We can no

longer get EXACTLY 5 "heads" in 10 tosses. It is obvious that the

previous events influenced the coin toss number 9.

A sequence of events means that the events do not take place at the

same time. They occur one after another.

The "Binomial Distribution Formula" shows some interesting facts. For

example, the probability to toss EXACTLY 1 "heads" in 10 tosses is

only 0.98%. It is quite difficult to get only 1 "heads" and 9 "tails"

in 10 tosses.

The probability to toss EXACTLY 5 "heads" in 10 tosses is 24.6%. It is

not that usual to get exactly 5 "heads" in 10 trials, even if the

individual chance of "heads" is 50%! We might have thought that we

would get quite often 5 "heads" and 5 "tails" in 10 coin tosses. NOT!

The chance is even slimmer to get 500 "heads" and 500 "tails" in 1000

tosses: 2.52%.

The probability to get 5 "heads" in 5 tosses represents, actually, the

probability of "5 heads in a row" (3.125%).

There is a data type limit. The number of trials N must not be larger

than 1500! There will be an overflow if you use very large numbers.

Blame the permutations and the limitations of the computers…

V. The function calculates the probability of at least M successes in

N trials for an event of probability p.

For example, we want to determine the probability of getting at least

4 heads in 10 tosses. Logically, the following situations qualify as

'success': 4 heads; 5 heads; 6 heads; 7 heads; 8 heads; 9 heads; and

10 heads. Obviously, the probability is better than the 'exactly 4 of

10' case.

There is a data type limit. The number of trials N must not be larger

than 1500! There will be an overflow if you use very large numbers.

Blame the permutations and the limitations of the computers…

VI. The function calculates the probability of at most M successes in

N trials for an event of probability p.

For example, we want to determine the probability of getting at most 4

heads in 10 tosses (no more than 4 in 10). In 'at least M in N' we

look at the glass as being half full. Why not look at it from the

pessimistic perspective: the glass can be empty sometimes (or present

degrees of emptiness)! Logically, the following situations qualify as

'success': 4 heads; 3 heads; 2 heads; 1 heads; and 0 heads. The

probability can be higher than the 'exactly 4 of 10' or 'at least 4 of

10' cases, but it won't be better from a player's perspective!

There is a data type limit. The number of trials N must not be larger

than 1500! There will be an overflow if you use very large numbers.

Blame the permutations and the limitations of the computers…

VII. The Probability to WIN AT LEAST 'K of M in P from N' at Lotto &

PowerBall

The official lotto odds are calculated as 'exactly K of M in P from

N'. For example, in a lotto 6/49 game, the player must play exactly 6

numbers per ticket. The lottery commission draws 6 winning numbers

from a field of 49. If the player plays only 6 numbers, the odds of

getting exactly 3 of 6 are 1 in 56.66. The player can play

combinations of 6 from a pool of 10 picks, for example. Now, the odds

can be calculated as exactly '3 of 6 in 10 from 49': 1 in 12.75.

In real life the player gets a better deal, however. The commission

does not oblige the players to 'exactly' situations. The real life

situation is 'at least K of M from N'. The commissions don't care if

you play just 6 numbers, or play a pool of picks. They don't care if

you expected 3 of 6 hits, but hit 4 of 6. They'll pay you for the

highest prize per ticket. It is clear that 'at least K of M from N' is

better than 'exactly K of M from N' from the player's perspective. If

the player plays 57 6-number random picks, the player should expect

one '3 of 6' hit. If playing 100 times 57 tickets, the expectation

should be 100 '3 of 6' hits. Sometimes, however, higher prizes can be

hit. That's why the odds of getting 'at least 3 of 6 from 49' are ' 1

in 53.66'.

Many lotto wheel aficionados might broadcast screams of happiness.

Cool down, Wheely! The previous calculations do not imply that 54

lines (combinations) will guarantee 100% in one draw a '3 of 6'

49-number wheel! Calculating the minimum number of successes for a

100% guarantee is a totally different matter. It is a book in itself

if one considers also the algorithm to generate the successes! Buy me

the best wine there is and we can talk about it. Otherwise wait for

the patents…

I wrote in the message "The Fundamental Formula of Wheeling":

"…the probability of winning [exactly] '3 of 6' is 1 in 57. FFG

calculates the median for p=1/57 as 39.16, rounded up to 40 for

DC=50%. How close is that figure (40) to reality in the UK 6/49

history file? The file I've been using for this analysis has 737 draws

(contains a few so-called 'extra' draws). The file has 733 regular

draws from the beginning of the game to the draw of January 1, 2003.

The median for the '3 of 6' case is around 38."

The calculations are correct, as far as the standard deviation is

concerned. The median, however is precisely in accordance with the

probability of at least '3 of 6': 1 in 54. It's right on the money!

Forget about one standard deviation! I had always noticed this small

discrepancy in all the data files I analyzed. The filter medians are

always a few points lower than the FFG medians. Now it fully makes

sense. The medians are the result of at least K of M in N probability,

NOT the exactly K of M in N probability!

We may consider from now on the Fundamental Formula of Gambling to be

the most precise instrument in games theory. There are a few posts at

this web site dealing with Markov chains: "Suspicion is mother of the

intellect; Markov chains".

Searching on Markov chains at Google yields close to 100,000 hits! The

topic is hot! I stated, however, that FFG outruns Markov by several

steps. Again, once and for all, the Fundamental Formula of Gambling is

be the most precise instrument in games theory. Unlike Markov chains,

FFG considers previous events to be of the essence for future events.

The events repeat precisely, according to the Fundamental Formula of

Gambling.

- continues ->

Ion Saliu