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. For 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 types of patterns to baccarat — and win BIG bucks… without knowing the formula!

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*.

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