Get ready to know the difference between Mutually exclusive and independent events and also to know what mutually exclusive and independent events are!
Two events let’s suppose event A and event B are said to be mutually exclusive if it is not possible that both of the events (A and B) occur at the same time.
For example, let’s consider the toss of a coin. When we toss a coin let A be the event that the coin lands on heads and let B be the event that the coin lands on tails.
In a single fair coin toss, events A and B are mutually exclusive which means the outcome can be either tails or heads. We cannot get both heads and tails at the same time.
Mutually exclusive events can be represented using a Venn diagram.
The following Venn diagram given below shows two mutually exclusive events A and B:
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If event A occurs, then there is no possibility of the occurrence of event B.
There are 52 Cards in a deck:
the probability of getting a King = 1/13, so we can say P(King)=1/13
the probability of getting a Queen is = 1/13, so we can say P(Queen)=1/13
When we combine those two Events, we cannot get queen and king at the same time thus,
P (A and B ) = 0
Therefore, we can say the probability of a King OR a Queen is (1/13) + (1/13) = 2/13
Events A and B are known as independent events if the probability of B occurring is unaffected by the occurrence of the event A happening
For example, let’s suppose that we are tossing a coin twice. Let A be the event that the first coin toss lands on heads and let B be the event that the second coin toss lands on heads.
Here, the Occurrence of event A does not affect event B in any manner.
Independent events can be represented using a Venn diagram.
The following Venn diagram given below shows two independents events A and B:
(Image to be added soon)
Probability of any event = Number of favorable outcomes / Total number of outcomes
For mutually exclusive events = P(A or B) which can also be written as P(A∪B)
And here P(A and B ) = 0
For independent events = P(A∩ B) = P(A).P(B)
At first the definitions of mutually exclusive events and independent events may sound similar to you. The biggest difference between the two types of events is that mutually exclusive basically means that if one event happens, then the other events cannot happen.
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Mutually exclusive and independent events
On the other hand, if the events are independent, then it means the occurrence and the outcome of any one event won’t have any effect on the occurrence and outcome of the other events.
Outcomes of rolling a die are mutually exclusive events. You can get either $5$ or $6$, but you can never get $5$ and $6$ at the same time.
Outcomes of rolling a die two times are independent events. The number we get on the first roll on the die has no effect on the number we’ll get when we roll the die one more time.
Mutually exclusive and independent events can be differentiated on the basis of Definition, Dependency, Occurrence of both events and Venn Diagrams.
Question 1. If we throw a dice twice, then find the probability of getting two 5’s.
Solution Let’s find the probability of getting 5’s,
The formula for finding the probability is,
Probability=Favorable outcomes/Total possible outcomes.
Total possible outcomes when we throw a dice are 6.
Probability of getting 5 on the first throw = 1/6
Probability of getting 5 on the second throw is also = 1/6
Let’s find the probability (Getting two 5’s), since they are independent events ,
Formula: P(A∩B) = P(A).P(B)
Probability of getting two 5’s = 1/6 ×1/6
Therefore, Probability of getting two 5’s = 1/6 ×1/6 = 1/36
1. Are Independent Events Always Mutually Exclusive and Are Mutually Exclusive Events Independent or Dependent?
No, mutually exclusive events (the events with non-zero probability) are always dependent. The definition of independence for events R and Q says that P(R and Q) = P(R) P (Q). However, if events P and Q are mutually exclusive, then P(R and Q) is equal to zero. If events R and Q are independent, then having information on event R does not tell us anything about event Q. If events R and Q are disjoint, then knowing that R occurs gives us information that event Q cannot occur. Disjoint or mutually exclusive events are always dependent since if one event occurs we already know the other one cannot happen.
2. What are Independent and Dependent Events and How Do You Know if an Event is Independent?
An event in which the outcome isn't affected by another event is known as an independent event. A dependent event is an event that is affected by the outcome of a second event. To test whether any two given events suppose A and B are independent, then we need calculate P(A), P(B), and P(A∩B), and then we need to check whether P(A∩B) equals P(A)P(B). If they are equal, then events A and B are independent; if not, the events A and B are dependent.