Types of Events in Probability

In daily conversations, people often use statements like “It might rain today” or “I will most probably pass the exam as the exam was not too tough” or “Most likely, he will be selected”. In all the 3 statements, words like might, probably, most likely are used, and they are used to indicate the probability or the certainty of something happening. So, if words can convey the probability of a certain event, why is an entire chapter dedicated to probability? It is because probability in math helps in determining the exact probability of an event to occur. For instance, there are 2 statements given- “It will probably rain today” and “there is a 70% chance of raining today”, which statement draws a better conclusion? The second statement since it tells a detailed probability of an event.

Probability

Probability of an event in mathematics is the prediction of that event to occur in numbers. Probability can be defined in a proportion that varies from 0 to 1, or it can also be expressed as a percentage varying from 0 to 100%. For example, there is a 0.8 percent chance of the meeting to postpone or there is an 80% chance of the meeting to postpone. The probability is always defined for the Events. Events can be of different types, Let’s learn about what they are and what are the types,

Events

Events in the most simple language are defined as the outcome of an experiment, when an experiment is done, some outcome is expected from the experiment and the expected outcome is known to be the Event in Probability. Every time the expected outcome will happen is not true, there is a chance that the event will occur, or it will not occur at all, the probability is actually the measurement of that occurrence of an event.

Sample space 

Sample space is defined as the set of all possible outcomes of the experiment and an event is one of the possible outcomes, likewise there can be more than one event (outcomes) of an experiment. Hence, it can be concluded that an Event is a Subset of Sample Space.

Types of Events

Since it is concluded that Events are the subsets of the Sample space, there will be one sample space for an experiment, but there can be multiple events of an experiment, it is important to note that events also have different types, let’s learn about them in detail,

• Independent Event

Independent events are those in which the next outcome is independent of the previous outcome. The means, the probability of the occurrence of an event will remain the same no matter how many times the same experiment is done. 

For instance, let’s take the example of rolling a die, a die is rolled once and the probability of getting an even number is 0.5, now the dice is rolled again, still the probability of getting an even number will be 0.5 only. This means, that the probability of the event is independent of its previous outcomes, such events are known as Independent events.

• 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.

For instance, let’s take the example of drawing balls from a bag, there are 4 black and 3 red balls in a bag, a ball is drawn, and it came out to be black (In the first draw, the probability of a black ball was 4/7= 0.571. When a ball is drawn the next time, the probability of the black ball to occur will change as now there are fewer balls in the bag (3 black and 3 red balls are left), hence, the probability of getting a black ball will be 3/6= 0.5. Such types of events are known as dependent events.

Note : In the example above, 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.

• Simple Event

Any event that comprises a single result from the sample space is known as a simple event. 

For instance, 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. 

For instance, S={1, 2, 3, 4, 5, 6} and E= {3, 4, 5}, here E is a Compound event.

• Mutually Exclusive Events

If the two events have nothing in common, then they are called mutually exclusive events, mutually exclusive events are similar to mutually exclusive sets.

For instance, S (sample space)= {23, 25, 27, 29, 31}, E₁= {23, 25, 27} and E₂= {29, 31}, as it is clearly seen that there is nothing common between the two sets, hence, events E₁ and E₂ are Mutually exclusive events.

• AND Event

AND Event is obtained by two or more than two events and the operation done on the events is the AND operation, 

For instance, E1 = {2, 3, 4, 5} and E2 = {3, 4, 7, 8}

E = E1∩ E2 = {3, 4}

• OR Event

OR Event is obtained by doing the OR Operation on two or more than two events.

For instance, E1 = {2, 3, 4, 5} and E2 = {3, 4, 5, 6}

E = E₁∪ E₂ = {2, 3, 4, 5, 6}

• Complementary Event

A complementary event is defined as the event which has the rest of the elements present in the sample space other than the event given. 

For instance, S = {1, 2, 3, 4, 5, 6, 7} and the given event E = {1, 2, 3}

The complementary event will be, E’ = {4, 5, 6, 7}

Sample Problems

Question 1: A die is thrown in the game of Ludo and E₁ denotes the event of getting even numbers and E₂ represents the event of getting a number more than 3, Find the Set for the following events,

1. E₁ or E₂
2. E₁ and E₂

Answer:

The sample space for the die will be,

S= {1, 2, 3, 4, 5, 6}

E₁ (only even numbers)= {2, 4, 6}

E₂ (number more than 3)= {4, 5, 6}

E₁ or E₂= {2, 4, 5, 6}

E₁ and E₂= {4, 6}

Question 2: A die is thrown and the set for the sample space obtained is, S = {1, 2, 3, 4, 5, 6}

E₁ is defined as the event of obtaining a number less than 5 and E₂ is defined as the event of obtaining a number more than 2.

Find the set for the following,

1. E₁ but not E₂
2. E₂ but not E₁

Solution:

Sample space will be, S= {1, 2, 3, 4, 5, 6}

E₁ (a number less than 5)= {1, 2, 3, 4}

E₂ (a number more than 2)= {3, 4, 5, 6}

1. E₁ but not E₂ = {1, 2}
2. E₂ but not E₁ = {5, 6}

Question 3: Write the sample space for tossing three coins at once, also answer the event of 2 exactly 2 heads at a time.

Answer:

When one coin is tossed at a time, the sample space is {H, T} since either head or tail can occur as a result. However, when three coins are tossed at the same time, the combination of different possibilities may happen, Those possibilities together will be composed of a sample space,

Tossing three coins, S= {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}

Hence, the sample space comprise 6 possible outcomes.

Event (E) for the occurrence of exactly two heads,

E= {(H, H, T), (H, T, H), (T, H, H)}

Question 4: Name the types of events obtained from the given below experiments,

1. A coin is tossed for the 5th time and the event of getting a tail when the first four times, the result was a head.
2. S (sample space)= {1, 2, 3, 4, 5} and E= {4}
3. S= {1, 2, 3, 4, 5} and E= {2, 4}
4. S= {1, 2, 3, 4, 5}, E₁= {1, 2} and E₂= {3, 4}

Answer:

1. No matter how many times the coin is tossed, every time the probability of getting a tail will be 0.5 irrespective of the previous outcomes, therefore the event will be an Independent event.
2. E= {4} is a Simple event.
3. E= {2, 4} is a compound event.
4. E₁ and E₂ are Mutually exclusive events.

Question 5: What are Impossible and sure events?

Answer:

Impossible events are those which are never to be occurred, they are the null sets and are indicated as {}. The probability of an impossible event to occur is 0. Hence, there are no outcomes seen.

While, sure events are nothing but the entire sample space since the probability of the event to occur in this case is 1.

Question 6: The sample space of an experiment is given as,

S = {10, 11, 12, 13, 14, 15, 16, 17} and the event, E is defined as all the even numbers. What will be the complementary event for E.

Answer:

S= {10, 11, 12, 13, 14, 15, 16, 17}

E (all even numbers) = {10, 12, 14, 16}

E’ (complementary of E)= {11, 13, 15, 17}