# 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 ˝ 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 ˝ 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

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