Content Strategy Manager
Mutually Exclusive is defined as a situation in which two different types of events cannot take place at the same time whereas independent events can be defined as the two different events that can take place at a time and out of these two one stays unaffected while the other event is happening. The set of all outcomes of tossing coins that give results either head or a tail but not both is an example of mutually exclusive events. And if two coins are tossed which have no effect on each other is an independent event.
Read More: NCERT Solutions for Class 12 Mathematics Chapter 13 Probability
Key Words: Mean, Variables, Probability, Statistics, Variance, Average value, Arithmetic mean, Events, Exclusive events, Independent events
What is an Event?
[Click Here for Sample Questions]
Event is a term that is defined as some actions or instances that happen at a particular place during a particular period. In other words, we can say that the results or outcome and the subsets of the outcomes which we get during an instance are known as events. For example, If we throw a dice, we will get some outcomes i.e. 1,2,3,4,5,6 where each of these outcomes is known as events.
Events
Also Read: Independent Events in Probability
Types of Events in Probability
[Click Here for Sample Questions]
There are different Types of Events that occur during probability. Some of which are given below
- Simple Events: An event containing only a single element of the sample space is called a simple event. These are also known as elementary events. For example, If we throw a dice, there is only one possibility of 2 appearing on the dice. This is a good example of a simple event and it is given by E={2}.
- Compound Events: This is just the opposite of a simple event in which we get more than one sample point on a sample space. These are also known as composite or mixed events. For example, If we throw a dice, the probability of getting an odd number will be more than one i.e. E=(3, 5)
Compound Events
- Independent Events: If the two events occur at the same time and any of the one events is unaffected by the happening of the other one is known as independent events.
- Dependent Events: The types of events that are surely affected by the occurrence of other events. These are called dependent events.
Mutually exclusive and Independent events
- Mutually Exclusive Events: Mutually exclusive events are defined as the exclusion of one event just by the occurrence of the other event.
- Exhaustive Events: When all the events share the total sample space, it is known as exhaustive events.
- Complementary Events: This can be defined as the event in which one event occurs if and only if the other does not. In other words, there is only one occurrence of an event when the other one will not occur. The probability of getting complementary events is only 1.
Also Read:
Mutually Exclusive and Independent Events
[Click Here for Sample Questions]
Mutually Exclusive Events are such types of events that cannot occur at the same time. They require separate times for the occurrence. For example, when we toss a single coin then there is only one chance or probability of getting either head or tail.
P(A∪B) = P(A) + P(B)
Here the symbol U stands for 'union' which means to take the elements from set A Or set B. Henceforth, the probability of occurrence of either A or B events, when A and B are mutually exclusive events is equal to the probability of occurrence of A plus the probability of occurrence of B.
Mutually Exclusive Event
Also Read: Multiplication Theory on Probability
Independent Events are those events that are independent of each other. It means that the occurrence of event A doesn't affect the occurrence of event B. For example, when we take two coins separately and toss them at once, then the result will come differently because both are separate. One event does not have any effect on other events.
P(A∩B) = P(A) · P(B)
We can use the equation to check if events are independent. Thus we have to multiply the probabilities of the two events together to see if they are equal to the probability of happening at the same time.
Independent Events
The video below explains this:
Independent Events Detailed Video Explanation:
Also Read: Maxima and Minima
Difference Between Mutually Exclusive Events and Independent Events
[Click Here for Sample Questions]
The key differences between Mutually Exclusive Events and Independent Events are as follows.
| Mutually Exclusive Events | Independent Events |
|---|---|
| Events that cannot happen at the same time and the one event will get affected by the occurrence of another event | In this, the occurrence of one event does not affect the happening of another event |
| There is a chance of getting affected by the other event or one event will get influenced by other events. | In this process, both events can occur freely or it will not have any influence on the happening of the other. |
| The mathematical expression of mutually exclusive events is P(XandY) = 0. | Independent events can be calculated by P(XandY) = P(X) P(Y) |
| Example: the outcomes of rolling a die | Example: The outcomes of rolling two die |
Read More: Relation Between Mean Median and Mode
Things to Remember
- Events are defined as every subset of a sample space is called an event. Various events happen due to some causes.
- There are various types of events like simple events, compound events, independent events, dependent events, mutually exclusive events, exhaustive events, complementary events.
- Mutually exclusive events are those events which cannot occur at the same time. There is the exclusion of other events due to the occurrence of the first event.
- Independent events are those events where two different events can take place at a time and out of these two one stays unaffected while the other event is happening.
Also Read: Difference Between Mean and Median
Sample Questions
Ques. A boy rolled two well-balanced dice and wrote down the results on a piece of paper. Determine the sample space. (3 Marks)
Ans. When two dice are rolled, there are (6×6)=36 outcomes. The outcomes are given below in the form of sets.
S = {(1, 1),(2,1),(3,1),(4,1),(5,1),(6,1),
(1, 2),(2,2),(3,2),(4,2),(5,2),(6,2),
(1, 3), (2,3),(3,3),(4, 3),(5, 3),(6,3),
(1, 4),(2,4),(3,4), (4,4), (5,4), (6,4),
(1, 5),(2,5), (3,5), (4,5),(5,5), (6,5),
(1, 6),(2,6) (3,6), (4,6), (5,6), (6,6)}.
Ques. Define the term Events. (2 Marks)
Ans. Event is a term in probability which means that every subset of a sample space is called an event.
Example, If we throw a dice once, we get
Sample space S= {1, 2,3,4,5,6}
Hence the event to get a prime is as given below.
E={2, 3,5}
Therefore, E is a subset of S.
Ques. A boy starts tossing a fair coin. List its sample space. (2 Marks)
Ans. In tossing a fair coin, there are two possible outcomes, namely
Head(H) and Tail(T).
Therefore, the sample space of the aforementioned question is given underneath.
S={H, T}.
Ques. A dice is thrown by someone. List its sample space. (2 Marks)
Ans. When we throw a dice, it can result in any of the six numbers.
Thus, 1,2,3,4,5,6.
Therefore, the sample space is S
S= {1,2,3,4,5,6}.
Ques. From a group of 3 boys and 2 girls, we select two children. What would be the sample space for this experiment? (2 Marks)
Ans. Let us name the boys as B1, B2 and B3, and the girls as G1 and G2. The sample space is as follows.
S = { B1B2, B1B3,B1G1, B1G2, B2B3, B2G1,B2G2, B3G1, B3G2, G1G2 }.
Ques. A die is thrown only if a coin is thrown twice and the second throw results in a tail. Describe the sample space. (2 Marks)
Ans. The sample space is as follows.
S={TH,HH, HT1,HT2,HT3,HT4,HT5,HT6, TT1, TT2, TT3,TT4,TT5, TT6}
H stands for head.
T stands for tail.
Ques. A coin is tossed once. Find the probability of getting a head. (2 Marks)
Ans. When a coin is tossed once, the sample space is given by S ={ H, T }.
E to be let as the event of getting a head.
Then, E = {H}
Therefore n(E) =1 and n(S) =2
P(getting a head) = P(E) = n(E)/n(S) = ½ .
Ques. What is the formula for 'probability of an event'? (2 Marks)
Ans. P(E)=number of outcomes favourable to the occurrence of E /number of all possible outcomes
P(E) = number of unique elements in E /number of unique elements in S
P(E) = n(E) / n(S)
Therefore, P(E)= n(E) /n(S).
Ques. A box contains cards numbered 6 to 50. A card is drawn at random from the box. Calculate the probability that the drawn card has a number which is a perfect square. (CBSE 2013)
Ans. Total number of cards = 50 – 6 + 1 = 45
Perfect square numbers are 9, 16, 25, 36, 49, i.e.,
5 numbers
∴ P(a perfect square) = 5/ 45 = 1/ 9
Ques. Cards marked with numbers 3, 4, 5, …, 50 are placed in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the selected card bears a perfect square number. (CBSE 2016)
Ans. Total no. of cards = 50 – 3 + 1 = 48
Perfect square number cards are 4, 9, 16, 25, 36, 49 i.e., 6 cards
∴ P(perfect square number) = 6/ 48 = 1/ 8
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments