# How to calculate probability of multiple events

Probability is the measure of the likelihood that an event will occur. In simple terms, it tells us how likely it is for a particular outcome to take place. When there are multiple events, calculating the probability of each event can become a bit tricky. However, with a few basic mathematical concepts and formulas, we can easily make sense of calculating the probability of multiple events.

In this article, we will explore various methods for probability calculation, such as independent and dependent events, combining probabilities using rules of addition and multiplication, and conditional probability.

**1. Independent Events:**

Independent events are those where the occurrence of one does not affect the other. For example, tossing a coin or rolling a dice multiple times. To calculate the probability of independent events occurring simultaneously, you can use the Multiplication Rule.

Multiplication Rule:

P(A and B) = P(A) × P(B)

For example, if we flip a coin twice, what’s the probability of getting heads both times?

P(Heads) = 1/2 (assuming a fair coin)

P(Heads twice) = P(Heads) × P(Heads) = (1/2) × (1/2) = 1/4

**2. Dependent Events:**

Dependent events are those where the occurrence of one event directly influences the other. For instance, picking cards from a deck without replacement. The probabilities will change as you pick more cards from the deck.

In such cases, we can use conditional probability to establish a relationship between the probabilities of both events happening.

Conditional Probability Formula:

P(A|B) = P(A ∩ B) / P(B)

This formula states that Probability of A given that B has occurred equals Probability of A intersection B divided by Probability of B.

Example: In a shuffled deck containing 52 cards with 4 queens, what is the probability of drawing two queens in a row without replacement?

P(Q1) = 4/52 (probability of drawing first queen)

P(Q2|Q1) = 3/51 (probability of drawing second queen after drawing first one)

P(Q1 ∩ Q2) = P(Q1) × P(Q2|Q1) = (4/52) × (3/51) = 1/221

**3. Combining Probabilities Using Addition Rule:**

To find the probability of either of the multiple events occurring, you can use the Addition Rule.

Addition Rule:

P(A or B) = P(A) + P(B) – P(A ∩ B)

This formula is used when we want to find the probability of either event A or event B occurring and taking into account any overlap between the two events.

Example: If A has a probability of 0.4, and B has a probability of 0.5, with their intersection having P(A ∩ B) = 0.2, what’s the probability that either A or B will occur?

P(A or B) = P(A) + P(B) – P(A ∩ B)

= 0.4+0.5-0.2

= 0.7

**Conclusion:**

Calculating the probability of multiple events may seem complex at first, but with the right formulas and understanding of independent and dependent events, you can easily tackle these problems. The key is to remember that different rules apply depending on whether events are independent or dependent, and whether you want to calculate the probability of both events happening at once or just one occurring. Keeping these basic concepts in mind will allow you to confidently solve various probability-based problems in your daily life and beyond.