An event is a particular outcome or a collection of outcomes from the sample space (the set of all possible outcomes) that we focus on or are interested in.
- The sample space (S) includes all possible outcomes.
- An event (E) includes one or more outcomes from that sample space that meet certain conditions.
Also, events can be classified into various types based on different properties and probability values of events.
It is now clear that events are subsets of the sample space. The following list gives the different types of events:
Impossible Event
These kinds of events are impossible and can be described by an empty set ∅ and are called Impossible Events.
Example:
Consider an experiment in which we roll a die. Now let's define an event that consists of outcomes that are multiple of 7. Sample space for this event is denoted by S,
S = {1, 2, 3, 4, 5, 6}
Now since there is no outcome in the sample space which is a multiple of 7. So, the set of event E will be an empty set.
Sure Event
Such an event, which has a probability of 1, i.e., occurrence of the event is certain or universal truth then that event is called Sure Event or Certain Event.
Note: An event that consists of the entire sample space is called a sure event (or certain event).
Example:
If we roll a die, as the event is the occurrence of a number less than 7, then it is sure that the occurring number is always less than 7 as the die only has numbers 1, 2, 3, 4, 5 and 6.
Simple Event
Any event that comprises a single result from the sample space is known as a simple event.
Example:
The Sample space of rolling a die, S = {1, 2, 3, 4, 5, 6} and the event for getting less than 2, E = {1}, where E has a single result taken from the sample space, Hence the event is a Simple event.
Compound Event
A Compound event is just opposite to what a simple event is, that is, any event that comprises more than a single result or more than a single point from the sample space, that event is known as a Compound event.
Example:
S = {1, 2, 3, 4, 5, 6} and E = {3, 4, 5}, where E is a Compound event.
Dependent Events
Dependent events are those in which the next outcome depends on the previous outcomes, which means, the probability of an event will change based on its previous outcomes.
Example:
Experiment: "Drawing balls from a bag consisting of 4 black and 3 red balls"
Event: "Drawing a black ball from the bag without replacement"
1. First draw.
The probability of drawing a black ball = 4/7= 0.571.2. After the first draw, (if a black ball was drawn)
Now the bag consist of 3 black and 3 red balls.
Now the probability of getting black ball is 3/6 = 0.5Thus, this event is dependent as the probability of each successive event depends upon the previous event.
Note : Here, there is a way of converting this dependent event into independent event, it can be done through Replacement. If after each experiment the ball is again kept in the bag, the sample space of the experiment will not change and hence, the probability of the event will remain same too.
Independent Event
Independent events are those in which the next outcome is independent of the previous outcome. This means the probability of the occurrence of an event will remain the same no matter how many times the same experiment is done.
Example:
Experiment: Rolling a die,
Event: "getting an even number"
1. Rolling the die : P(getting an even number) = 0.5,
2. Now the dice is rolled again, still P(getting an even number) = 0.5.
This means, that the probability of the event is independent of its previous outcomes.
Equally Likely Events
Those outcomes of an experiment that have the same probability are called Equally Likely Events. In other words, if two or more events have the same likelihood of happening, they are considered equally likely events.
Example:
Consider rolling a fair six-sided die. Each of the six possible outcomes (1, 2, 3, 4, 5, and 6) has the same probability of occurring, which is 1/6.
Therefore, rolling a 1 is equally likely as rolling a 2, 3, 4, 5, or 6.
Mutually Exclusive Events
Mutually exclusive events are events that cannot happen at the same time. If one happens, the other cannot.
Example:
When you roll a die:
- Event A: Getting a 2.
- Event B: Getting a 5.
These two events are mutually exclusive because you cannot get both 2 and 5 on the same roll.
Exhaustive Events
Exhaustive events are a set of events that cover all possible outcomes of an experiment. This means at least one of them must happen.
Example:
Tossing a coin:
- Event A: Getting Head.
- Event B: Getting Tail.
These two events are exhaustive because together they include all possible outcomes of the coin toss.
Problems on Types of Events
Problems 1: Two coins are tossed simultaneously. Define the events:
- A : Getting at least one head
- B : Getting no head.
- Are A and B mutually exclusive?
Solution:
Sample space,
S = {HH, HT, TH, TT}
A = {HH, HT, TH}
B = {TT}
A ∩ B = ∅
Since their intersection is an empty set, A and B are mutually exclusive events.
Problems 2: A die is rolled twice. Let event A be “sum of numbers obtained is 7” and event B be “first die shows 4.” Find A ∩ B.
Solution:
Sample space (not listed fully due to size) contains all ordered pairs (x, y) where x and y ∈ {1, 2, 3, 4, 5, 6}.
For event A:
A = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
For event B:
B = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)}
Now,
A ∩ B = {(4,3)}
Hence, A ∩ B = {(4,3)} — only one outcome satisfies both events.
Problems 3: A card is drawn from a standard deck of 52 cards. Let event A be “drawing a king” and event B be “drawing a red card.” Find A ∩ B and A ∪ B.
Solution:
Total cards = 52
Event A: King cards = {K♠, K♣, K♥, K♦}
Event B: Red cards = {All ♥ and ♦ = 26 cards}
A ∩ B = {K♥, K♦}
A ∪ B = {All red cards (26) + black kings (2)}
Hence, number of outcomes in A ∪ B = 26 + 2 = 28.
Problems 4: Two dice are rolled.
- A: “Sum of numbers is even”
- B: “Both numbers are even.”
- Are these events mutually exclusive?
Solution:
Sample space has 36 outcomes.
For event A: sum even ⇒ {(1,1), (1,3), (1,5), (2,2), (2,4), (2,6), …} (all outcomes where sum = even)
For event B: both numbers even ⇒ {(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)}
Now,
A ∩ B = {(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)}
Since intersection is not empty, A and B are not mutually exclusive events.