Published on January 27, 2001. *O tempora! O mores!*

**FORMULA** has been upgraded to calculate also the * Binomial Distribution Formula (BDF)*. This formula calculates the probability of

*Sequence of events* means that the events do not take place at the same time: They occur *serially*, one after another.

The * Binomial Distribution Formula* shows some interesting facts. For example, the probability to toss EXACTLY 1

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 is 50%! We might have thought that we would get quite often 5 *heads* and 5 *tails* in 10 coin tosses. After all, it is a 50-50 proposition. 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* (

We can work with complicated case scenarios as well. Let's take, as an example, **roulette**. (Methinks you're here mainly because of a strong interest in gambling mathematics.)

- The roulette probabilities
*p*of*Red*(*R*) and*Black*(*B*) are equal:*p=18/37*. What is the chance of this 4-spin pattern:*B-R-B-R*or*R-B-R-B*? - Calculate the degree of certainty for
**EXACTLY**2 successes in 4 trials (spins):*37.4%*or*1 in 2.7* - There are 2^4 possible
*B-R*outcomes:*16* - The final chance for one
*B-R-B-R*pattern:*0.374 / 16 = 0.023 = 1 in 43*spins. - The
*R-B-R-B*pattern has the same degree of certainty:*1 in 43*spins. - It happens quite frequently in
*1 to 1*gambling games. The Asians “secretly” apply these— and win BIG bucks… without knowing the formula!**types of patterns in baccarat**

The software has a data **size** limit. The number of trials **N** must not be larger than **1500**. *N!* (*N factorial*) for numbers larger than 1500 leads to impossible-to-handle amounts for today's PCs. There will be an overflow if you use very large numbers...

Axiomatic one, here is the generalized formula for * exactly M successes in N trials*:

**BDF** = probability, chance of * exactly M successes in N trials*;

- If we can calculate the probability for multiple successes as
**exactly**, we can also calculate probabilities as**at least**and as**at most**. Instead of formulas (one-step calculations), we apply algorithms (multiple-step calculations). -
**at least**M successes: add up the probabilities for*M*,*M+1*,*M+2*, etc. ... up to*N*. -
**at most**M successes: add up the probabilities for*0*,*1*,*2*, etc. ... up to*M inclusively*.

- Axiomatic one, the 16-bit Formula program was superseded (or greatly upgraded) to 32-bit software that runs on all 32-bit/64-bit
*Windows*versions, including*Windows 10*. Application name:**SuperFormula**(see screenshot below). - The
is in the function**Binomial Distribution Formula***E =*.**Exactly**M Successes in N Trials - I upgraded this formula even further by adding two more useful functions:
*L =*and**At Least**M Successes in N Trials*M =*.**At Most**M Successes in N Trials - As we saw previously, the chance to toss
**exactly**5*heads*in 10 tosses is 24.6%. - The chance to get
**at least**5*heads*in 10 tosses is higher, of course: 62.3%. - The chance to get
**at most**5*heads*in 10 tosses is higher, of course: 62.3%. - The
**at least**and**at most**formulas are together convincing proof of the fallacy of: Long losing streaks are probabilistically-equal to long winning streaks. Casinos and many gambling "authors" want to convince you otherwise, especially when trying to combat Ion Saliu's gambling theory. They want gamblers to see only long losses so that you would abandon all mathematical systems.**gambler's fallacy**

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