Binomial Distribution, Binomial, Distribution, Formula, Software, Free.

Binomial distribution formula, probability, past, number of trials

Binomial Distribution, Binomial, Distribution, Formula, Calculate.

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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