What are Independent Events?

Suppose you flip a coin twice. When you first flip the coin; whether it is heads or tails, it will not influence the second time the coin is flipped. Each flip is independent and stands on its own. This is the essence of independent events: two or more events where outcome of one event has no impact on other event. Even in our day-to-day life, independent events can be seen everywhere. IIT JAM exam and IISER exam prefer that you understand this concept practically since conceptual questions are asked in the exam. For example:

• When you first roll a die and get a 3, then, you roll the die again and you get a 5.
• You draw a card from the deck, then, putt it back, shuffle it, and then, draw again.
• Your chance to win a lottery ticket does not change if your neighbor buys a lottery ticket as well.

Mathematically, two events, A and B, will be independent if:

P(A and B) = P(A) × P(B)

This means the probability of both events happening together is the product of their individual probabilities.

Let us take a look at the process to understand how to determine if both events are independent of each other since it is important for NEET exam and the JEE Main exam:

• Check for the definition: First, you need to determine whether the outcome of one event affects the other event? If the answer is no, they are most likely independent.

For example, we will toss a coin and roll a die. The result of this coin toss will not impact the outcome of the roll of die.

• Use Probability Rules: Let us calculate P(A), P(B), and P(A and B). If P(A and B) equals to P(A) × P(B), then, events are independent.

Let us suppose you have drawn a card from the deck. You will note it, put it back, and the draw the card again. The probability of drawing a King twice, the second time is still 4/52, even it was drawn the first draw.

• Real-World Context: Think about the cause and effect. If one event does not influence the other, they are independent.

For example, the chance of rain today and chance of winning a board game tonight are independent since these are unrelated.

Keep a note, if events involve replacement such as drawing a card and putting it back, these are often independent events. Without replacement, they usually are not independent.

Mutually exclusive events are different events. These events cannot occur at the same time. If one event occurs, the other cannot. Let us roll a die and get a two or a five in a single roll. You cannot get both numbers at once. The key difference between independent events and mutually exclusive events is in their occurrence. Independent events can happen together, but mutually exclusive events cannot happen together.

Mathematically, when two events are mutually exclusive, they will be:

P(A and B) = 0

They will also follow:

P(A or B) = P(A) + P(B)

The following table explains the difference between independent events and dependent events:

Parameters	Independent Events	Dependent Events
Impact	One event does not impact the other.	One event will influence the other.
Example	Flipping a coin two times.	Drawing two cards from the deck without replacing the first card that was drawn.
Formula	P(A and B) = P(A) × P(B)	P(A and B)=P(A)×P(B∣A)
Usecase	Events with replacement or no connection.	In scenarios without replacement or with any direct link.