If two events are mutually exclusive, what is the probability that both occur at the same time?

Probability has its basis in a lot of real-world applications. Based on past outcomes and the number of turns, as well as other data available, you can derive the probability a cricket team has of winning the toss. Similarly, you can also calculate the probability of several day-to-day things if you know concepts like events and conditional probability. 

Here, we will take you through the basic concepts, and from there, help you understand the practical approach towards our question. 

Introduction to Events

When you conduct a random experiment, the outcome is known as an event. In our earlier example, getting heads or tails can be defined as an event during the toss. 

The total number of outcomes for a particular experiment is known as sample space. If you roll a dice, there is a chance of rolling 1, 2, 3, 4, 5, and 6. All six possibilities are part of the sample space.

There are various aspects of events that you need to know to solve conditional probability questions. 

#1 The Union of Events

When we take a combined approach to understand the probability of a particular event, it is known as the union of events. For instance, event A is when you get a 4 on your dice, while event B is when you get a 5 on the dice. he probability of AUB is: 

P(C)= P(AUB)

#2 Intersection of Events

The second condition you ought to be aware of when you are deriving probability is the events’ intersection. Let’s take the previous question; A being the event when you get 4 on the dice and B being the event when you get 6 on the dice. The probability of getting multiples of these numbers is:

P(C) = P (AB)

#3 Disjoint Events

When two events cannot occur together at any point in time, they are called disjoint events. If you roll a die and don’t get 3 and 5 together,  it is called a disjoint event. 

Event A = Getting multiple of 3

Event B = Getting a multiple of 5

Event A = {3,6}

Event B = {5}

Sample Space = {1,2,3,4,5,6}

There is no way the two events can occur together. So, they are disjoint events. 

Let’s talk about another set of events that you should know.

The Exclusive, Independent & Dependent Events

Independent Events

When event A does not disturb event B, they are said to be independent events. For instance, the outcome of a toss will not impact the outcome of rolling a die.  Similarly, tossing the coin, rolling the die, and choosing a marble from a jar are three completely different events. Neither has a role in the other event. They are all independent. 

The probability of independent events is calculated as follows:

P (AB) = P(A) *P(B)

Dependent Events

If your second event’s outcome is dependent on the first event, then it is a dependent event. For instance, if you roll a die and get five, what is the probability that when you roll the dice again, you will get another five? The outcome of the first event determines the probability of the second event. 

Mutually Exclusive and Exhaustive Events

When two events don’t happen together, they are called mutually exclusive events. They are neither dependent, nor can they happen at the same time. For instance, you can get a 1, 2, 3, 4, 5, or 6 when rolling a die. All six outcomes are mutually exclusive for a particular event. Similarly, when you toss a coin, you will either get heads or tails. You cannot possibly get both at the same time. These outcomes are known as mutually exclusive.

The exhaustive events are a set of events, which contain all the possible outcomes for the event. The sample space, in many cases, is equivalent to the set of exhaustive events.

Conditional Probability

Conditional probability is a natural consequence of your experiments. The outcome of a particular experiment may impact other experiments as well. In this case, you will derive the outcome for event B, under the condition that event A has already occurred. The outcome of event A will determine the course event B will take. For instance, you have already taken an ace of cards from the deck. The probability that you can get another ace depends on the first event.

Event A = Event that has occurred 

Event B= Event we are calculating 

Conditional Probability P(B/A) = P(A and B)/P(A)

So, now we have all the formulas and understand events. Let’s work on the main concept here. 

If two events are mutually exclusive, what is the probability that both occur at the same time?

As we have already discussed, mutually exclusive events occur independent of each other and never occur together.

As such, the probability will be zero for this case.

You will know the events are mutually exclusive if, the following is true:

P(A/B) = P(A)

P(B/A)= P(B)

P(A and B) = P(A)*P(B)

There are two possibilities again in this case, when you are talking about two different events. For instance, the events occur with replacement and without replacement. 

With replacement: The member of the sample space is replaced before the occurrence of the second event. For instance, you pick one card out of the 52 cards in the deck. Before you move onto the next event, you put that card back in the deck. This is called sampling with replacement.

Without replacement: In this case, you pick the card but, you don’t put it back in the deck before you shuffle it again. This means the second event is completely independent of the first event. There is a good chance that the outcome of the first event may not be the outcome of the second event. 

Let’s solve some problems to get the hang of probability and understand conditional events. 

Sample questions:

1. The research group collected data for yearly road accidents. There were two conditions:
2. People who followed the traffic rules
3. People who did not observe any rules

The group is interested in knowing the probability of road accidents caused by people who followed traffic rules differently:

Traffic Rules Followed

No Rules Followed

Accident

50

500

No Accident

2000

5000

A= People who followed the traffic rules

B= Accident

P(c) = P(A+B)/P(A)

= 50/2050

= 0.0243

2. A coin is tossed three times. A is the event when you get at least two heads, while B is the event when you get at least two tails. Find the P(A/B).

A={HHH,HTH,THH,HHT}; B={TTT,TTH,HTT,THT,HHT,THH,HTH}

P(A?B)=3 and P(B)=7

P(A/B)=P(B)P(A?B)?

P(A/B)=73?

Let’s take a look at a few more instances in probability. 

1. The Probability of Equally Likely Events

If the outcome of a trial is equally likely, then they are called equally likely events. For instance, there is an equal chance of getting heads and tails at the end of the experiment. So, the event is equally likely.

There is an equally likely chance of getting 1, 2, 3, 4, 5, or 6 on the die after rolling it. 

2. Probability of Random Events

An unpredictable event wherein you are not aware of the outcome is a random event. For instance, predicting the probability of you falling off a height is impossible. It is a random event that you may not have thought of. Similarly, you can never be sure of the number of supporters in a particular campaign. The probability of people supporting a cause from a random group is unpredictable. However, there is still a way in which you can find the probability of a random event.

Let’s say 100 out of 700 people are supporting a cause. You just identified 4 people randomly, and you want to know if they support the cause or not. 

100 people from a rough 700 are supporting the cause. 100/700=0.142

In this case, you have picked 4 people randomly. So, you need to multiply this decimal four times. 0.142*0.142*0.142*0.142

The answer will determine the event’s probability. 

Conclusion

Probability is an important concept in the real world.  It gives you the certainty of the number of outcomes and how it can change a scenario. It is especially important for data scientists, as it gives them the first step towards constructing an algorithm. From understanding what is data to building a detailed training model, Cuemath offers a complete understanding to build your career as a data modeling scientist. To learn more about our educational programs and identify the one that suits you most, you can connect with our tutors and specialists.

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