The theory of probability

  Introduction
  Foundation
  Calculating
  Presize definition
  Binomial distribution
  Hypergeometric distribution
  Combined events
  Inseparable events
  'Ion Saliu's Paradox'
  Combinatorics
  Markov Chains
  Random Walks
  Contacts

As long and complicated as it may be sometimes, calculating the probability p is only the first step! If we determine that the probability p equals 1 in something, it does NOT guarantee that the event will take place in a number of trials equal to something. You flip the coin two times. The probability to get heads is 1 in 2. You expect one heads, but don't hold your breath! The degree of certainty, DC is 75% that heads will come up in 2 tosses. In a significant percentage (25% of cases), heads will not show up. That's far from negligible.

The degree of certainty can be viewed as a probability of probability strongly connected to a number of trials. The master formula that calculates the number of trials N for an event of probability p to appear with a degree of certainty DC is known as the Fundamental Formula of Gambling. We may also call it the Fundamental Formula of The Universe or the Formula of The Everything.

If p = 1 / N, we can discover an interesting relation between the degree of certainty DC and the number of trials N. The degree of certainty has a limit, when N tends to infinity. Let's analyze a few particular cases.

• Rolling the unbiased dice; actually just one die. The probability to get any one of the point faces is p = 1/6. The degree of certainty DC to get any one of point faces in 6 throws is 66.5%.

• Spinning the roulette wheel. The probability to get any one of the 38 numbers is p = 1/38. The degree of certainty DC to get any one of the numbers in 38 spins is 63.7%.

• Let's look at a case with a very large number of possibilities, therefore a very low probability — a lotto 6/49 game. Total possible combinations in a 6/49 lotto game is 13,983,816. The probability to get any one of the combinations is p = 1/13,983,816. The degree of certainty DC to get any one of the numbers in 13,983,816 drawings is 63.212057% (0.63212057).

• The limit of the degree of certainty DC is {1 — (1/e)} when N tends to infinity,
for an event of probability p = 1/N and a number of trials equal to N.
e represents the base of the natural logarithm and equals approximately 2.718281828...
The limit {1 — (1/e)} is approximately 0.63212055...

• Let it be forever known as the Ion Saliu's paradox (problem) of N trials.

In the so-called Problem of Coincidences or Couple (Spouse) Swapping we had encountered a very similar limit: 1/e. Very interesting how everything in life, indeed in the Universe is...coupled!

• Is it the same difference — the same thing gambling this way? For example, play one lotto ticket N consecutive draws, or N tickets in one drawing? Something is very clear now. If you play one roulette number for 38 spins, you are not guaranteed to win! You have only a 63.7% chance to win. On the other hand, if you play all 38 numbers, you are guaranteed to win! You, who have ears to hear, don't bet it all on one spin or number. Play more numbers or tickets at once. The probability is significantly lower for two or more numbers or combinations to have simultaneously long losing streaks...