Binomial distribution formula, probability, past, number of trials

Analysis of the Fundamental Formula of Gambling, probability, odds, standard deviation, binomial distribution, house edge, payouts. The past matters in random events: Number of trials.

Posted by Ion Saliu on August 05, 2000.

In Reply to: Is it important what happened in past lotto drawings? posted by Steve Cochrane on July 28, 2000.

• This is the most divisive issue in theory of gambling. Does the past matter or not in random events? Isn’t the probability of getting “Heads” always ½ or 50%? Yes, it is. At the same time, it is true that the probability to get 100 “Heads” in a row is much, much lower than tossing one “Heads” only. The key point here is we are dealing with two different events.

There is a comprehensive formula named the “binomial distribution” or the “formula of M successes in N repeated events”. Let’s consider p the probability of an event. The term p is always constant from trial to trial. For example, the probability of “Heads” in coin tossing is always ½ or 50%; it never changes. Let’s consider q the probability that the event does not occur (e.g. “Heads” does not occur). In this case q = 1 – p (an elementary rule in theory of probability). Then, the probability of exactly M successes in N trials is:

N! / ((N – M)! x M!) x (p to the power of M) x (q to the power of (N – M))

(I have no better editor to write this formula!)
The ! sign represents the factorial of a number. N! = 1 x 2 x 3 x … x (N –1) x N
For example, 3! = 1 x 2 x 3 = 6 (We can arrange 3 elements A, B, C in a total of 6 ways: ABC, ACB, BAC, BCA, CAB, CBA).

The “formula of M successes in N repeated events” has two particular cases.
1) If M=1 and N=1: “one appearance in one trial”; the result of formula is p (the individual probability of the event, e.g. to get one “Heads” in one toss).
2) If M = N : “N appearances in N trials”; equivalent to “N consecutive appearances”.

If the event is viewed as a sequence of trials, we may say that the “past” matters in random events. That’s why we reach absurd situations such as “100 heads in a row is equal to one heads at a time”. We can ruin all the casinos in the world if they implemented a game like the following. A coin is tossed. The casino wins only after three “Heads” in a row. The player wins after one “Heads” or two “Heads” in a row. All other events are “ties” or “pushes”. Has anyone heard of such a game?

Ion Saliu

Statistics theory as I learned it says that in an honest game every combination has equal probability, each time regardless of what happened before. If thats true why wouldn't a play of lotto combination 1,2,3,4,5,6 be as likely an outcome as anything generated by the lotto software programs?

Ion Saliu

Resources in Theory of Probability, Mathematics, Statistics, Standard Deviation, Software

• The Best Introduction to Standard Deviation: Theory, Algorithm, Software.

• The Latest On: Standard Deviation, Variance, Variability, Fluctuation, Volatility, Variation, Dispersion, Median, Mean Average.
• Standard Deviation, Gauss, Normal, Binomial, Distribution. Calculate: Median, degree of certainty, standard deviation, binomial, hypergeometric, average, sums, probabilities, odds.

• Generate combinations inside the bell (Gauss) curve, around the median.
• Upgrade to FORMULA: standard deviation, binomial distribution.
• Probability, odds, standard deviation, binomial software.
• Software and formulae to calculate lotto odds using the hypergeometric distribution probability.
• Primer: Probability, Odds, Formulae, Algorithm, Software Calculator.

• Software: Science, Mathematics, Statistics, Lexicographic, Combinatorial.

Comments:
: Binomial Distribution Formula Software. This is the most dividing issue in theory of gambling. Does the past matter or not in random events? Isn’t the probability of getting “Heads” always ½ or 50%? It is true that the probability to get 100 “Heads” in a row is much, much lower than tossing one “Heads” only. The key point here is we are dealing with two different events. : There is a comprehensive formula named the “binomial distribution” or the “formula of M successes in N repeated events”. Let’s consider p the probability of an event. The term p is always constant from trial to trial. For example, the probability of “Heads” in coin tossing is always ½ or 50%; it never changes. Let’s consider q the probability that the event does not occur (e.g. “Heads” does not occur). In this case q = 1 – p (an elementary rule in theory of probability). Then, the probability of exactly M successes in N trials is: The “formula of M successes in N repeated events” has two particular cases. : 1) If M=1 and N=1: “one appearance in one trial”; the result of formula is p (the individual probability of the event, e.g. to get one “Heads” in one toss). : 2) If M = N : “N appearances in N trials”; equivalent to “N consecutive appearances”. : If the event is viewed as a sequence of trials, we may say that the “past” matters in random events. That’s why we reach absurd situations such as “100 heads in a row is equal to one heads at a time”. We can ruin all the casinos in the world if they implemented a game like the following. A coin is tossed. The casino wins only after three “Heads” in a row. The player wins after one “Heads” or two “Heads” in a row. Analysis of "Binomial Distribution, Binomial, Distribution, Formula, Software, Free." I have read through the mathmatical justifications for the Lotto systems posted here and can't say I understand them completely. The thing that I really don't get is - what difference does it make what happened in past lotto drawings? Statistics theory as I learned it says that in an honest game every combination has equal probability, each time regardless of what happened before. If thats true why wouldn't a play of 1,2,3,4,5,6 be as likely an outcome as anything generated by the lotto software programs? The key point here is we are dealing with two different events. Analyze, analyse Binomial Distribution, Binomial, Distribution, Formula, Software, Free at www.saliu.com. There is a comprehensive formula named the “binomial distribution” or the “formula of M successes in N repeated events”. Let’s consider p the probability of an event. The term p is always constant from trial to trial. For example, the probability of “Heads” in coin tossing is always ½ or 50%; it never changes. Let’s consider q the probability that the event does not occur (e.g. “Heads” does not occur). In this case q = 1 – p (an elementary rule in theory of probability). Then, the probability of exactly M successes in N trials is: N! / ((N – M)! x M!) x (p to the power of M) x (q to the power of (N – M)). The “formula of M successes in N repeated events” has two particular cases. 1) If M=1 and N=1: “one appearance in one trial”; the result of formula is p (the individual probability of the event, e.g. to get one “Heads” in one toss). If the event is viewed as a sequence of trials, we may say that the “past” matters in random events. That’s why we reach absurd situations such as “100 heads in a row is equal to one heads at a time”. We can ruin all the casinos in the world if they implemented a game like the following. A coin is tossed. The casino wins only after three “Heads” in a row. The player wins after one “Heads” or two “Heads” in a row. All other events are “ties” or “pushes”. Has anyone heard of such a game? FORMULA: software to perform sophisticated gambling and probability calculations: degree of certainty standard deviation binomial distribution. FORMULA software standard deviation politics successes trials lotto odds binomial average norm normal distribution formula probability number software mathematics theory of probability gambling formula system. Statistics drawings coin coin toss standard deviation 3 SD 3 standard deviations analyze drawings method numbers winning combinations random chance previous event normal probability. " FORMULA - calculate fundamental formula of gambling, degree of certainty, number of trials, binomial distribution formula, binomial standard deviation. Title" "Binomial distribution formula, probability, past, number of trials". Article on the Fundamental Formula of Gambling, probability, odds, standard deviation, binomial distribution, house edge, payouts. The past matters in random events: Number of trials.