Probability

Mutually Exclusive Events

Out of a collection of events, if at a given time the occurrence of only one of them is possible. Then, that collection is a collection of mutually exclusive events. For example, the event of a person being an adult. The person is either an adult or not an adult, no in between. So, these two events are mutually exclusive.

Suggested Videos

Play
Play
Play
previous arrow
next arrow
previous arrownext arrow
Slider

 

Mutually Exclusive Events

If the occurrence of an event makes the occurrence of another event impossible for that particular experiment then they are mutually exclusive. It becomes an ‘either-or’ situation.

  • For example, the outcome after rolling a die can be either an even number or an odd number.
  • So, {1, 3, 5} and {2, 4, 6} are sets of mutually exclusive events.

Browse more Topics under Probability

Mutually Exclusive and Exhaustive Events

The condition for mutually exclusive events for being exhaustive is the outcome of an experiment must be one out of the sample space of mutually exclusive events for that particular experiment.

For example, the blood group of a person. The events are {A, B, AB, O}. These are the only possible blood groups and a person can have only one of these. So, we can call blood test as an experiment and all the possible outcomes as mutually exclusive and exhaustive eventsThese events are equally likely as well.

Definition

In case of the events being mutually exclusive, exhaustive and equally likey the probability is

P(A) = mA/m      where,

  • P(A) is the probability of A
  • mA is the no. mutually exclusive, exhaustive and equally likely events favourable to event A.
  • m is the total no. of mutually exclusive, exhaustive and equally likely events.

The above-given definition is the classical definition of probability. It is applicable only when the events are finite and equally likely. Taking set theory into consideration we can write the definition of probability as:

Exclusive

Here,

  • S = sample space of finite no. elementary events for an experiment.
  • A = A ⊂ S,  A is an event under consideration then,
  • P(A) = n(A)/n(S)   where,
  • n(S) denotes the total no. of events in S.
  • n(A) denotes the no. of outcomes favourable to A.

Now, complement of A(i. e., A’) will be: n(A’) = n(S) – n(A).

⇒n(A’)+ n(A) = n(S)

A and A’ make a pair of mutually exclusive and exhaustive events. This gives, for any two mutually exclusive events say A and B.

Exclusive

  • A ∩ B = Φ ⇒ n(A ∩ B) = 0
  • ⇒P(A ∩ B) = 0; 0 is the minimum possible which is the probability of an impossible event.
  • (A ∩ B) is an impossible event because they are exclusive and t is impossible for them to occur simultaneously.
  • P(A ∪ B) = P(A) + P(B)

For mutually exclusive and exhaustive events P(A) + P(B) = 1 because

  • The maximum possible probability is 1, which is the probability of a sure event.
  • Since P(A) and P(B) are exhaustive they are the only two events.
  • ∴ P(A ∪ B) = 1; (A ∪ B) is a sure event as one of the two events are sure to occur for the experiment.

Solved Examples on Mutually Exclusive Events

Question: An urn contains balls of various colours. The probability of drawing red is 1/3, blue is 1/7, green is 1/9. If a ball is drawn at random what is the probability that

  1. Either red or blue is drawn
  2. Neither red nor blue is drawn
  3. None out of red, blue, green gets drawn

Solution. a) It is given that, P(R) = 1/3 and P(B) = 1/7
∴ P(R∪B) = P(R) + P(B)
= 1/3 + 1/7 = 10/21
All three events R, B and G are exclusive and that’s why their individual probabilities are added

b) P(R∪B)’ = 1 – P(R∪B)
= 1 – 10/21 = 11/21
Notice here that since the events R, B, and G are exclusive but not exhaustive P(R∪B)≠P(G).

c) P(R∪B∪G)’ = 1 – P(R∪B∪G)
= 1 – (1/3 + 1/7 + 1/9)
= 1 – 29/45 = 16/45
As the sets are not exhaustive P(R∪B∪G)’≠0.

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

3
Leave a Reply

avatar
2 Comment threads
1 Thread replies
0 Followers
 
Most reacted comment
Hottest comment thread
2 Comment authors
Mandeep P ShettyMandeepKumar Recent comment authors
  Subscribe  
newest oldest most voted
Notify of
Kumar
Guest
Kumar

There are so many errors in two of the lectures that I have watched. The flow of the lectures are also inappropriate. Firstly you never defined what an event is. For this lecture you can just say that an event is a subset of sample space. Therefore it can be any subset of sample space, even phi(empty set) or the whole sample space itself. You are confusing events with elements of sample space. There is a fundamental errors on tis page too. Like P(A|B) is probability of event A given that event B has (already) occurred. However the text in… Read more »

Mandeep P Shetty
Guest
Mandeep P Shetty

@Kumar Thank you! I thought I was somehow wrong in my understanding. Another error is: “Even compound events (two events occurring at the same time) can be independent events. Ex. Tossing a coin and rolling a die. Sample space S = {(1,H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.” The example is still simple even and not a compound event. An actual example for a compound event will be ex: Tossing a coin and rolling a die and getting a 1 every time. The sample… Read more »

Mandeep
Guest
Mandeep

@Kumar Thank you! I thought I was somehow wrong in my understanding. Another error is: “Even compound events (two events occurring at the same time) can be independent events. Ex. Tossing a coin and rolling a die. Sample space S = {(1,H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.” The example is still simple even and not a compound event. An actual example for a compound event will be ex: Tossing a coin and rolling a die and getting a 1 every time. The sample… Read more »

Customize your course in 30 seconds

Which class are you in?
No thanks.