🧮 How to Calculate Probabilities
Introduction
Calculating probabilities is a fundamental skill in statistics and data analysis. This practical tutorial presents step-by-step methods to calculate probabilities in different scenarios, from simple situations to more complex cases involving lotteries and combined events.
What is Probability?
Probability is a numerical measure that quantifies the chance of an event occurring. It ranges from 0 (impossible event) to 1 (certain event), or can be expressed as a percentage from 0% to 100%.
Basic Formula
P(Event) = Number of favorable outcomes / Total number of possible outcomes
This is the fundamental formula of probability, also known as classical probability.
Example 1: Simple Probability
Let's start with a basic example: what is the probability of getting "heads" when flipping a coin?
Solution:
- • Possible outcomes: {heads, tails} = 2 outcomes
- • Favorable outcomes (heads): 1 outcome
- • P(heads) = 1/2 = 0.5 = 50%
Example 2: Probability with Dice
What is the probability of getting an even number when rolling a 6-sided die?
Solution:
- • Possible outcomes: {1, 2, 3, 4, 5, 6} = 6 outcomes
- • Favorable outcomes (even numbers): {2, 4, 6} = 3 outcomes
- • P(even number) = 3/6 = 1/2 = 0.5 = 50%
Probability of Combined Events
When we have multiple events, we need to consider how they relate:
Independent Events
Two events are independent when the occurrence of one does not affect the probability of the other.
Formula:
P(A and B) = P(A) × P(B)
Example: What is the probability of getting "heads" twice in a row?
P(heads and heads) = P(heads) × P(heads) = 0.5 × 0.5 = 0.25 = 25%
Mutually Exclusive Events
Two events are mutually exclusive when they cannot occur simultaneously.
Formula:
P(A or B) = P(A) + P(B)
Example: What is the probability of getting 1 or 6 on a die?
P(1 or 6) = P(1) + P(6) = 1/6 + 1/6 = 2/6 = 1/3 ≈ 33.33%
Probability in Lotteries
Calculating probabilities in lotteries involves combinations. Let's use Mega-Sena as an example:
Mega-Sena: Matching 6 Numbers
In Mega-Sena, you choose 6 numbers from a total of 60. Order does not matter.
The total number of possible combinations is calculated by:
C(60,6) = 60! / (6! × 54!) = 50,063,860
Therefore, the probability of matching all 6 numbers with a single bet is:
P(6 matches) = 1 / 50,063,860 ≈ 0.000002%
Mega-Sena: Matching 4 Numbers (Quadra)
To match 4 numbers, you need to:
- • Match 4 of the 6 drawn numbers
- • Miss 2 numbers (choosing from the 54 non-drawn numbers)
- • C(6,4) × C(54,2) = 15 × 1,431 = 21,465 favorable combinations
- • P(4 matches) = 21,465 / 50,063,860 ≈ 0.043%
Step by Step: Practical Calculation
Follow these steps to calculate any probability:
- 1. Identify the sample space: List all possible outcomes.
- 2. Identify favorable outcomes: Determine which outcomes correspond to the event of interest.
- 3. Apply the formula: Divide the number of favorable outcomes by the total number of outcomes.
- 4. Simplify: Reduce the fraction to its simplest form.
- 5. Convert (optional): Express as decimal or percentage as needed.
Practical Exercises
Exercise 1:
In an urn with 10 balls (3 red, 4 blue and 3 green), what is the probability of drawing a red ball?
View solution
P(red) = 3/10 = 0.3 = 30%
Exercise 2:
When rolling two dice, what is the probability of getting a sum of 7?
View solution
Favorable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes
Total outcomes: 6 × 6 = 36
P(sum = 7) = 6/36 = 1/6 ≈ 16.67%
Important Tips
- • Use fractions when possible: Fractions are more accurate than decimals.
- • Check your answer: Probability is always between 0 and 1 (or 0% and 100%).
- • Beware of dependent events: When events are dependent, the probability of the second event changes after the first.
- • Practice with real examples: Apply the concepts in everyday situations for better understanding.
Conclusion
Calculating probabilities is an essential skill that combines mathematical logic with practical understanding of random events. Mastering the basic concepts presented in this tutorial allows you to analyze everyday situations and make more informed decisions based on probabilistic data.
Next steps: Explore our articles on Probabilities and Combinations and Theoretical Probabilities to deepen your knowledge.