Often Wrong on the Issue of Opportunity? Here's How to Understand Coincidental Events

Illustration of the probability of a compound event (freepik)

YOGYAKARTA - The possibility of a compound event often makes many students frustrated when faced with exams. In fact, this concept is actually easy to understand if you know the formula and how to apply it correctly.

Many people mistakenly distinguish between independent and independent events. As a result, the calculation results are far off. For this reason, let's learn a precise technique so that we don't make a mistake again!

The notion of joint probability

Quoted from the e-Modul Mathematics Book for XII Class of the Ministry of Education and Culture, the joint probability is the probability of two or more events occurring at once in an experiment.

Unlike simple events that only involve one outcome, compound events combine multiple possible outcomes. For example, when throwing two dice at once or taking several cards from a set of bridge cards.

In everyday life, an example is the probability of a soccer team winning two consecutive matches. This compound event is a subset of a sample space more complex than a single event.

Types of Mutually Exclusive Events Independent Events (Independent Events)

Two events are said to be independent if the occurrence of the first event does not affect the occurrence of the second. For example, the simultaneous throwing of two dice. The formula for the probability is: P(A ∩ B) = P(A) × P(B)

The event of mutually exclusive (mutually exclusive)

The event of mutual independence is two events that cannot occur simultaneously. For example, the appearance of the numbers 2 and 5 in one dice throw. The formula for the probability: P(A ∪ B) = P(A) + P(B)

The incident did not pass each other.

Two events that still have the possibility of occurring simultaneously. For example, taking a heart card or a King card from a set of bridge cards. The formula for the probability is: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

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Formula for the probability of a joint event

Well, in calculating the probability of a compound event, there are three main formulas that need to be mastered, including the following:

First, for independent events use the formula P(A ∩ B) = P(A) × P(B), that is, when two events do not affect each other. Second, independent events use the formula P(A ∪ B) = P(A) + P(B) because the two events are unlikely to occur simultaneously. Third, for non-independent events, the formula is P(A ∪ B) = P(A) + P(B) - P(A ∩ B) because there is a possibility of an intersection between the two events. Note the keyword "and" for the intersection operation and "or" for the union operation, and always check whether there is an intersection of events before choosing the correct formula. Problem Example and Discussion Example 1: Independent Events

A die and a coin are thrown simultaneously. What is the probability of getting a 4 on the die AND a picture on the coin?

Discussion:

P(number 4) = 1/6

P(image) = 1/2

P(number 4 ∩ image) = 1/6 × 1/2 = 1/12

Example 2: The Event of Mutual Release

From a set of bridge cards, what is the probability of picking a heart card OR a spade card?

Discussion:

P(heart) = 13/52 = 1/4

P(spade) = 13/52 = 1/4

Because they are mutually exclusive: P(hearts ∪ spades) = 1/4 + 1/4 = 1/2

Example 3: Non-Mutual Release

What is the probability of taking a heart card OR a King card?

Discussion:

P(heart) = 13/52

P(King) = 4/52

P(heart ∩ King) = 1/52 English: P(heart ∩ King) = 1/52

P(heart ∪ King) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13

Tips Solving Problems How to quickly distinguish the type of event

Create a tree diagram for complex experiments such as inspection of defective (C) or non-defective (T) products. Then use a Venn diagram to see the intersection of events

Keyword identification: "and" means multiplication, "or" means addition

Common mistakes to avoid

Don't forget to reduce the slices in the event of not being separated. Then make sure the sample space is correct before calculating the probability.

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