How is the sum rule applied in probability?
Probability is a fascinating and essential concept in the understanding of uncertainty and randomness. In exploring the mathematical rules behind possible events, we come across the sum rule, a key principle for calculating the probability of occurrence of one or another event. This content aims to guide you through the mechanism of the sum rule and allow you to recognize its usefulness in real-life scenarios.
What is the sample space and why is it important?
Before delving into the addition rule, it is essential to understand the concept of the sample space: the set of all possible outcomes of a random experiment. Suppose we roll two fair dice. Here, each die has six sides, implying that there would be 36 possible outcomes (6 sides of the first die times 6 sides of the second die).
Exploring this sample space allows us to identify:
- Individual outcomes: Each combination of throws of two dice.
- Individual probabilities: The probability of each of those outcomes.
How to calculate probabilities using union and intersection?
Now, let's think about calculating the probability of obtaining a specific result when throwing the dice. For example, the probability that at least one of the dice is a '1'. Exploring our sample space, we determine that there are 11 possible combinations where at least one '1' is obtained. This translates to a probability of ( \frac{11}{36} ).
Expanding this example, we could also calculate the probability of getting an even sum when rolling both dice. Since there are 18 possible combinations that add up to an even number, its probability is ( \frac{18}{36} ).
How do you use the addition rule?
At this point, you might think that adding the two probabilities above is enough to find the probability of getting a '1' or an even sum. However, this is where the intersection comes in: some outcomes meet both conditions (such as double '1').
The sum rule formula tells us:
[ P(A \cup B) = P(A) + P(B) - P(A \cap B) ]
Applying this, 5 events counted twice (their intersection in the Venn diagram) are subtracted, setting the probability correctly to ( \frac{24}{36} ) or ( \frac{2}{3} ).
What about independent events?
What if we want to calculate the probability that the sum of the dice is even or odd? In this case, the sum cannot be simultaneously even and odd. If two events are mutually exclusive (as here), the intersection is zero. So, the probability of getting an even or odd sum is simply the sum of their individual probabilities.
An exercise to challenge you
Imagine that we survey 100 people about their country of origin and their favorite sport. As an exercise, determine:
- What is the probability that one participant is from Latin America?
- What is the probability that he/she prefers basketball?
- Based on the above answers: What is the probability that a participant is from the United States or prefers sports other than soccer or basketball?
Combine your calculations, use the addition rule and set logic to solve this challenge - leave me your feedback and trust that you can master the addition rule with practice!
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