🎲 Theoretical Probabilities
Introduction
Theoretical probabilities are grounded in rigorous mathematical principles and provide the basis for understanding the real chances of different events occurring. When we play the lottery, we often ask: 'What is the chance of winning?' Theoretical probabilities give us precise and objective answers to these questions, based purely on mathematics, without depending on strategies or promised systems.
This study explores the mathematical calculations behind probabilities in lottery games. Understanding these probabilities is fundamental to having realistic expectations and making informed decisions about games of chance. Theoretical probabilities do not change - they are fixed and determined by the mathematical rules of the game.
Combinatorics Fundamentals
Combinatorics is the area of mathematics that studies the different ways of arranging, combining and selecting elements. It is fundamental for calculating probabilities in lottery games, as it allows us to count how many different combinations are possible.
Think of it this way: if you need to choose 6 numbers from 60, how many different ways exist to make that choice? Combinatorics gives us the tools to answer this question precisely and systematically.
Combination Formula
Combination Formula
C(n,r) = n! / (r! × (n-r)!)Where:
- • n: n: Total elements available (e.g., 60 numbers in Mega-Sena)
- • r: r: Number of elements chosen (e.g., 6 numbers per bet)
- • ! ! (factorial): Product of all integers from 1 to n (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120)
Why does this formula work? The formula counts all possible ways of choosing r elements from n, disregarding order. The factorial in the denominator eliminates the different orders of the same combination (since {1, 2, 3} equals {3, 2, 1} in a combination).
Calculations by Lottery
Let's calculate theoretical probabilities for the main Brazilian lotteries:
Mega-Sena
Configuration: 6 numbers from 1 to 60
C(60,6) = 60! / (6! × 54!)
C(60,6) = 50,063,860
Probabilities:
- • Sena: 1/50,063,860
- • Quina: 1/154,518
- • Quadra: 1/2,332
Percentages:
- • Sena: 0.000002%
- • Quina: 0.000647%
- • Quadra: 0.0429%
Quina
Configuration: 5 numbers from 1 to 80
C(80,5) = 80! / (5! × 75!)
C(80,5) = 24,040,016
Probabilities:
- • Quina: 1/24,040,016
- • Quadra: 1/64,106
- • Terno: 1/866
Percentages:
- • Quina: 0.000004%
- • Quadra: 0.00156%
- • Terno: 0.115%
Lotofácil
Configuration: 15 numbers from 1 to 25
C(25,15) = 25! / (15! × 10!)
C(25,15) = 3,268,760
Probabilities:
- • 15 hits: 1/3,268,760
- • 14 hits: 1/21,791
- • 13 hits: 1/691
Percentages:
- • 15 hits: 0.000031%
- • 14 hits: 0.00459%
- • 13 hits: 0.145%
Probability Comparison
To contextualize these probabilities, let's compare them with everyday events:
Comparison with Real Events
Everyday Events:
- • Struck by lightning: 1/1,000,000
- • Win state lottery: 1/10,000
- • Die in plane crash: 1/11,000,000
- • Win at bingo: 1/100
Lotteries:
- • Mega-Sena (Sena): 1/50,063,860
- • Quina (Quina): 1/24,040,016
- • Lotofácil (15): 1/3,268,760
- • Timemania (Sena): 1/25,827,165
Conditional Probabilities
Conditional probabilities are important for understanding more complex scenarios:
Example: Probability of Not Winning
Mega-Sena: Probability of Not Hitting Any Number
To not hit any number, you need to choose 6 numbers that were NOT drawn.
C(54,6) / C(60,6) = 30,260,340 / 50,063,860
≈ 0.6046 or 60.46%
This means that in 60% of cases, you will not hit any number.
Law of Large Numbers
The Law of Large Numbers tells us that as the number of attempts increases, the observed frequency approaches the theoretical probability.
🎯 Practical Implications
- • In many attempts, results approach theoretical probabilities
- • Temporal deviations are normal and expected
- • In the long run, the 'law of averages' applies
- • Each individual attempt maintains the same probability
Expected Value
Expected value is an important measure in probability analysis:
Example: Expected Value in Mega-Sena
If the Mega-Sena prize is R$ 100,000,000 and you bet R$ 4.50:
Expected Value = (100,000,000 × 1/50,063,860) - 4.50
Expected Value = 1.997 - 4.50 = -2.50
On average, you lose R$ 2.50 per bet.
Conclusions
Theoretical probabilities provide a solid mathematical basis for understanding real chances in lottery games. They show us that:
- The chances of winning are extremely low
- Each draw is independent of previous ones
- There are no strategies that change the probabilities
- Expected value is always negative for the player
⚠️ Important
Understanding theoretical probabilities is fundamental for a conscious and responsible approach to lottery games. The chances are mathematical and cannot be changed by strategies or promised 'systems'.