Understanding Probability Through Intuition
In this article, we’ll explore the concept of probability in an intuitive manner, using real-life examples to demystify probability theory.
What is Probability?
Probability measures the chance of an event happening, scaled from 0 to 1, where 0 means the event is impossible, and 1 means the event is certain. While this definition seems straightforward, the nuances of probability extend into different types, each with its unique application and interpretation.
Types of Probability
Theoretical (Exact) Probability
For example, if you flip a fair coin, there are two possible outcomes — heads or tails. So, the exact probability of getting heads is 1 (one way it can happen) divided by 2 (two possible outcomes), which is 0.5 or 50%.
Exact probability is determined theoretically based on the nature of the situation; hence, its application in real-life scenarios is limited.
It is defined as the number of ways a specific event can occur divided by the total number of equally likely outcomes. This type of probability is based on theoretical reasoning rather than actual experiments.
Empirical Probability
Every day, I watch how often my pet parrot drinks water. After a week of observation, I have the following data:
- Number of days that my parrot drinks water only once a day = 2
- Number of days that my parrot drinks more than once a day = 5
- Total days of observation = 2 + 5 = 7
So, based on this historical data, I can say that for the next day, there is a ~28% (2/7) probability that my parrot will drink water only once for the whole day and a ~71% (5/7) that he will consume water more than once. Such probability is calculated based on past data and, more specifically, the frequency of observed events. Remember that empirical probability heavily depends on the quality and quantity of the observation data on which it’s based, meaning that it is just an approximation of the actual likelihood of the event. In summary, the empirical probability is determined by conducting experiments and collecting data; hence, it has many applications for real-world scenarios.
Subjective Probability
“I estimate a 35% chance Barcelona wins the next Champions League.”
Subjective probability expresses a personal belief in an event’s occurrence, influenced by individual judgment, experience, or available information. This probability type underscores the variability of belief-based estimations across different observers.
Independence vs. Dependence
In some situations, the outcome of one event can affect the likelihood of another event happening. In these cases, the events are said to be dependent. If the outcome of one event doesn’t affect the likelihood of another event happening, they are said to be independent. Let’s examine the following events:
- Event A: going to the office by car.
- Event B: winning the lottery.
- Event C: having a car crash.
Speaking empirically, events A and C are dependent, and events A and B are independent.
Events Occurring Together
Imagine you have a standard six-sided die and a coin. You want to find the probability of rolling a three and then flipping heads. These are two separate events:
- Rolling a three on the die (Event A): The probability is P(A) = 1/6 since there is one favourable outcome (rolling a 3) out of six possible outcomes.
- Flipping heads with the coin (Event B): The probability is P(B) = 1/2, since there are two outcomes (heads or tails) and one is favourable (heads).
Since these two events are independent (the result of the die roll doesn’t affect the coin flip), you can calculate the combined probability by multiplying the probabilities of each event: P(A and B)=P(A) × P(B)
Grouping events
Imagine you have a bowl containing 5 apples, 3 oranges and 2 pineapples, and you randomly select one piece of fruit from the bowl.
- Event A: You pick an apple.
- Event B: You pick an orange.
The probabilities are:
- P(A) = 5/10 because there are 5 apples out of 10 total pieces of fruit.
- P(B) = 3/10 because there are 3 oranges.
Since these events are mutually exclusive (you can’t pick an apple and an orange at the same time with one choice), you can add the probabilities to find the likelihood of picking either an apple or an orange: P(A or B) = P(A) + P(B) = 8/10
Conditioning
Imagine you’re planning to play street football with friends, and the enjoyment of your game depends heavily on the weather. You’re particularly concerned about rain.
Let’s define two events:
- Event A: It rains on the day of the street football game.
- Event B: The street football game is enjoyable.
You want to assess the probability that the street football game will be enjoyable, given that it does not rain. This probability is referred to as Conditional Probability and is notated as P(B|A’) where A’ is the event where “It doesn’t rain on the day of the street football game”.
Probability Theory: Building Blocks
- Experiment: A process generating outcomes, such as flipping a coin.
- Sample Space: All possible outcomes of an experiment. A coin toss’s sample space is {Heads, Tails}.
- Event: A subset of the sample space, like rolling an even number {2, 4, 6} on a die.
- Probability: The likelihood of an event, ranging from 0 (impossible) to 1 (certain).
- Random Variable: Assigns numerical values to outcomes. For a die showing three, a random variable X = 3.
- Probability Distribution: Shows probabilities across a random variable’s values. A fair die gives each outcome 1/6 chance.
- Expected Value: The mean outcome over many trials. A fair die’s expected value is 3.5, signifying the average roll over time.
- Independence: Events are independent if one’s outcome doesn’t affect the other.
- Conditional Probability: The probability an event occurs given another event has already happened.
Axioms of Probability Theory
Probability theory is built on foundational axioms:
- Non-negativity: P(A) ≥ 0 for any event A in the sample space S.
- Normality: P(S) = 1, confirming that some outcome in the sample space must occur.
- Additivity: For mutually exclusive events, their union’s probability equals their probabilities’ sum.
This article has aimed to intuitively explore probability theory, focusing on real-life examples rather than abstract definitions. Hopefully, this approach has made the concepts more accessible and engaging.
