Contents

Two events are disjoint if they cannot happen simultaneously. That is, if one event happens, the other cannot.

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## Introduction

In probability theory and statistics, two events are said to be disjoint or mutually exclusive if they cannot occur simultaneously. This means that if one event happens, then the other cannot happen. If two events are not disjoint, then they are said to be joint or overlapping.

### What are events?

Events are things that happen. They can be big, like an earthquake, or small, like a cat meowing. They can be ordinary, like a sunrise, or unusual, like a solar eclipse.

Events can be physical events, like a ball being thrown, or mental events, like a memory being recalled. They can be observable events, like a car stopping at a red light, or private events, like a thought occurring in someone’s mind.

Events can also be classified according to whether they are happens concurrently (at the same time) or sequentially (one after the other).

Concurrent events are two or more events that take place at the same time. For example, if two people are talking on the phone at the same time, they are engaging in concurrent phone calls. If three people are all talking at the same time, they are engaging in concurrent conversations.

Sequential events are two or more events that take place one after the other. For example, if you first brush your teeth and then eat breakfast, you are engaging in sequential activities.

### What are disjoint events?

Two events are said to be disjoint if they cannot happen at the same time. In other words, if one event happens, the other cannot. For example, imagine you are flipping a coin. The two possible outcomes are heads or tails. If you get heads, then you know for certain that you will not get tails (and vice versa). This is because it is impossible to get both heads and tails from a single coin flip – they are disjoint events.

Another way of thinking about it is that the probability of two disjoint events happening is simply the sum of the probabilities of each event happening individually. For example, the probability of flipping a coin and getting either heads or tails is ½ + ½ = 1 (or 100%).

## Examples of Disjoint Events

If two events cannot happen simultaneously, then they are disjoint events. For example, if you are rolling a dice, the event of getting a 1 and the event of getting a 2 are disjoint events. In other words, they cannot happen at the same time.

### Two dice are rolled

An event is disjoint if it cannot happen at the same time as another event. For example, two dice are rolled. The events “a 1 is rolled” and “a 2 is rolled” are disjoint because it’s impossible to roll a 1 and a 2 at the same time.

### A coin is tossed and a die is rolled

There are two possible outcomes when two events, such as tossing a coin and rolling a die, aren’t related. These outcomes are known as disjoint events.

For example, let’s say we have a bag containing five coins: two pennies, two dimes, and a nickel. We randomly select one coin from the bag and then flip it. The possible outcomes of this experiment are shown in the table below.

Flip Outcome Probability

Heads 2/5 or 0.4

Tails 2/5 or 0.4

Two Events That Are Disjoint 1/5 or 0.2

As you can see, there’s a 0.4 probability of flipping heads, a 0.4 probability of flipping tails, and a 0.2 probability of the events being disjoint (flipping heads and flipping tails).

## Properties of Disjoint Events

In probability theory and statistics, two events are said to be disjoint or mutually exclusive if they cannot occur simultaneously. This means that if one event happens, the other cannot. For example, the events “rolling a die and getting a 6” and “rolling a die and getting a 3” are disjoint because you cannot roll a die and get both a 6 and a 3.

### The probability of disjoint events is the sum of the probabilities of the individual events

Probability is the study of how likely it is for something to happen. In other words, probability helps us understand the likelihood of an event occurring. There are two types of events that can occur: disjoint and non-disjoint. Disjoint events are events that cannot happen at the same time, while non-disjoint events are events that can happen at the same time.

The probability of disjoint events is the sum of the probabilities of the individual events. For example, if you have a set of coins and you want to know the probability of flipping a coin and getting heads, you would add up the probabilities of flipping a coin and getting heads on each individual coin. This would be written as:

P( flipping a coin and getting heads ) = P( flipping a Coin A and getting heads ) + P( flipping a Coin B and getting heads ) + …

Non-disjoint events are different. The probability of non-disjoint events is not just the sum of the probabilities of each individual event, but also includes the probability that both events will happen at the same time. For example, if you have a set of coins and you want to know the probability of flipping a coin and getting tails, you would need to add up the probabilities of flipping a coin and getting tails on each individual coin, as well as the probability that both coins will land on tails at the same time. This would be written as:

P( flipping a coin and getting tails ) = P( flipping a Coin A and getting tails ) + P( flipping a Coin B and getting tails ) + … + P( Both coins landing on tails )

### The events are mutually exclusive

Disjoint events are two events that cannot happen at the same time. In other words, they are mutually exclusive. This means that if Event A happens, then Event B cannot happen, and vice versa. An easy way to remember this is that the word “disjoint” has the word “joint” in it, and two events that are disjoint cannot share a joint outcome.

For example, let’s say we’re flipping a coin. The possible outcomes are heads or tails (mutually exclusive). So if the result is heads, it cannot also be tails (and vice versa). Another example is picking a card from a standard deck of 52 cards. The suits are mutually exclusive (you cannot have a card that is both a spade and a heart), so if you pick a spade, it cannot also be a heart.

## Applications of Disjoint Events

In probability theory and statistics, two events are said to be disjoint or mutually exclusive if they cannot occur simultaneously. This means that if one event occurs, the other cannot. In other words, they are “disjointed” or ” mutually exclusive”. There are many applications of disjoint events, which we will explore in this article.

### Insurance

In insurance, two events are disjoint if they cannot happen at the same time. For example, you can either have car insurance or health insurance, but not both at the same time. This is because the events of having car insurance and having health insurance are disjoint.

### Gambling

Gambling is a perfect example of where disjoint events are very important. For example, when you bet on red at the roulette table, the events of you hitting black or green are disjoint with the event of you hitting red. This is because it is impossible for the ball to land on more than one color at a time.

Similarly, when you play blackjack, the events of being dealt a 10 or an ace are disjoint with the event of being dealt a 7. This is because you can only be dealt one card at a time and each card can only have one value. These types of disjoint events are extremely important in gambling as they allow casinos to calculate their odds and set their payouts accordingly.

## Conclusion

In conclusion, disjoint events are two events that cannot happen at the same time. They are mutually exclusive. In other words, if one event happens, the other cannot happen. An example of this would be getting a head and a tail when flipping a coin. You can either get a head or a tail, but not both at the same time.