# When Two Events are Disjoint, They are Also Independent

When two events are disjoint, they are also independent. This means that the probability of one event occurring does not affect the probability of the other event occurring.

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## Introduction

In statistics, two events are said to be disjoint if they cannot happen at the same time. For example, the events “drawing an ace from a deck of cards” and “drawing a king from a deck of cards” are disjoint. Similarly, the events “rolling a die and getting a 1” and “rolling a die and getting a 2” are also disjoint.

One important thing to note is that if two events are disjoint, then they are also independent. This is because the probability of one event happening has no bearing on whether or not the other event will happen.

To illustrate this, let’s say that you have a deck of cards that contains four Ace of Spades cards, four Ace of Hearts cards, four Ace of Clubs cards, and four Ace of Diamonds cards. This means that there is a 1/4 chance of drawing an Ace of Spades card, a 1/4 chance of drawing an Ace of Hearts card, a 1/4 chance of drawing an Ace
of Clubs card, and a 1/4 chance of drawing an Ace of Diamonds card.

Now, let’s say that you draw one card from the deck without replacement (this means that you don’t put the drawn card back into the deck before drawing again). What is the probability of drawing an Ace given that you’ve already drawn one?

Well, since there are only three Ace cards remaining in the deck (ACE♠️ , ACE♥️ , ACE♦️ ), and there are 52-3=49 total cards remaining in the deck, then we can see that the probability is 3/49.

## What are Disjoint Events?

Disjoint events are two events that cannot happen at the same time. In other words, the events are mutually exclusive. An example of disjoint events would be flipping a coin and getting heads OR flipping a coin and getting tails; these two events cannot occur simultaneously.

Mathematically, we can say that two events A and B are disjoint if the probability of them both occurring is zero, or P(A and B) = 0.

## What are Independent Events?

Two events are independent if the occurrence of one event does not affect the probability of the other event. In other words, the events are not related in any way.

For example, consider the two events “rolling a die” and “getting a number greater than 4.” These two events are independent because rolling a die does not affect whether or not you get a number greater than 4. The probability of each event is not affected by the other event.

Another way to think about independent events is that they are not affected by each other. For example, if you roll a die and get a 6, that does not affect the probability of getting a number greater than 4 on your next roll. The two events are independent.

## Disjoint Events are Independent Events

If two events are disjoint, then they are also independent. This is because the probability of one event happening does not affect the probability of the other event happening. For example, if you roll a die and get a 1, then the probability of getting a 2 on your next roll is 1/6. This is true whether or not you got a 1 on your first roll.

## Conclusion

We have seen that when two events are disjoint, they are also independent. This is because the probability of one event occurring does not affect the probability of the other event occurring. We can also say that two events are independent if the occurrence of one event does not affect the likelihood of the other event occurring.

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