Chapter 16.1 - Event in Probability

In the previous section, we completed a discussion on sample space. In this section, we will see event.

Event can be explained in 9 steps:
1. Consider the experiment of tossing a coin twice.
• We know that the sample space is:
S = {(H,H), (H,T), (T,H), (T,T)}
2. Suppose that, we are interested in those outcomes in which H occurs exactly once.
• Then we can pick out two outcomes: (H,T) and (T,H).
• We can form a set E using the two outcomes that we picked out:
E = {(H,T), (T,H)}
• We can write 3 points:
(i) We are interested in those outcomes in which H occurs exactly once.
• So we say that:
In the experiment, if H occurs exactly once, we have an event.
(ii) Two outcomes are favorable for the event.
• We write a set E in such a way that, those two outcomes are the only elements.
(iii) Then E will be a subset of S.
3. Suppose that, we are interested in those outcomes in which T occurs exactly two times.
• Then we can pick out one outcome: (T,T).
• We can form a set E using the outcome that we picked out:
E = {(T,T)}
• We can write 3 points:
(i) We are interested in those outcomes in which T occurs exactly two times.
• So we say that:
In the experiment, if T occurs exactly two times, we have an event.
(ii) One outcome is favorable for the event.
• We write a set E in such a way that, that one outcome is the only element.
(iii) Then E will be a subset of S.
4. Suppose that, we are interested in those outcomes in which T occurs at least once.
• Then we can pick out three outcomes: (H,T), (T,H) and (T,T).
• We can form a set E using the three outcomes that we picked out:
E = {(H,T), (T,H), (T,T)}
• We can write 3 points:
(i) We are interested in those outcomes in which T occurs at least once.
• So we say that:
In the experiment, if T occurs at least once, we have an event.
(ii) Three outcomes are favorable for the event.
• We write a set E in such a way that, those three outcomes are the only elements.
(iii) Then E will be a subset of S.
5. Suppose that, we are interested in those outcomes in which number of H is atmost 1.
("atmost 1" means, the number must not exceed 1. In other words, maximum allowed is 1)
• Then we can pick out three outcomes: (H,T), (T,H) and (T,T).
• We can form a set E using the three outcomes that we picked out:
E = {(H,T), (T,H), (T,T)}
• We can write 3 points:
(i) We are interested in those outcomes in which number of H is atmost 1.
• So we say that:
In the experiment, if H occurs atmost one time, we have an event.
(ii) Three outcomes are favorable for the event.
• We write a set E in such a way that, those three outcomes are the only elements.
(iii) Then E will be a subset of S.
6. Suppose that, we are interested in those outcomes in which second toss is not head.
• Then we can pick out two outcomes: (H,T) and (T,T).
• We can form a set E using the two outcomes that we picked out:
E = {(H,T), (T,T)}
• We can write 3 points:
(i) We are interested in those outcomes in which second toss is not H.
• So we say that:
In the experiment, if second toss is not H, we have an event.
(ii) Two outcomes are favorable for the event.
• We write a set E in such a way that, those two outcomes are the only elements.
(iii) Then E will be a subset of S.
7. Suppose that, we are interested in those outcomes in which number of T is atmost 2.
• Then we can pick out all the four outcomes: (H,H), (H,T), (T,H) and (T,T).
• We can form a set E using the four outcomes that we picked out:
E = {(H,H), (H,T), (T,H), (T,T)}
• We can write 3 points:
(i) We are interested in those outcomes in which number of T is atmost 2.
• So we say that:
In the experiment, if number of T is atmost 2, we have an event.
(ii) Four outcomes are favorable for the event.
• We write a set E in such a way that, those four outcomes are the only elements.
(iii) Then E will be a subset of S.
8. Suppose that, we are interested in those outcomes in which number of T is more than 2.
• Then we can pick out none of the four outcomes.
• We can form only a null set. A null set is also a subset of S.
• We can write 3 points:
(i) We are interested in those outcomes in which number of T is more than 2.
• So we say that:
In the experiment, if number of T is more than 2, we have an event.
(ii) No outcome is favorable for the event.
• We write a set E which is a null set.
(iii) Then E is a subset of S.
(Recall that, null set is also a subset)
9. Based on the above steps, we can write a definition for event. It can be written in 2 steps:
(i) An event is a set. It is denoted using the letter ‘E’
(ii) It is a subset of S.
• So all elements of E are outcomes.

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Occurrence of an event

This can be explained in 4 steps:
1. Consider the experiment of rolling a die.
• We know that S = {1, 2, 3, 4, 5, 6}
2. We are interested in those outcomes which are less than 4.
• Then we can pick out three outcomes: 1, 2 and 3.
• We can write a set E using the three outcomes that we picked out:
E = {1, 2, 3}
3. Now we can write about the occurrence of the event:
• When the die is rolled, if 1 is obtained, then we say:
Event E has occurred.
• When the die is rolled, if 2 is obtained, then we say:
Event E has occurred.
• When the die is rolled, if 3 is obtained, then we say:
Event E has occurred.
4. Based on the above three steps, we can write the definition for “occurrence of event”. It can be written in 3 steps:
(i) Let an outcome 𝛚 of an experiment occur.
(ii) Let 𝛚 be an element of E. In other words, 𝛚 ∈ E.
• Then we say that:
Event E has occurred.
(iii) If 𝛚 ∉ E, then we say that:
Event E has not occurred.

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Types of events

There are four types of events.
(i) Impossible event  (ii) Sure event  (iii) Simple event  (iv) Compound event.

Impossible event

This can be explained in 3 steps:
1. Consider the experiment of rolling a die.
• We know that S = {1, 2, 3, 4, 5, 6}
2. Suppose that, we are interested in those outcomes which are multiples of 7.
• Then we can pick out no outcomes.
• So the set E will be a null set.
3. If E is a null set, then that event is an impossible event.
• We can do the experiment any number of times we like. We will never get an outcome which is an element of E. Because, E is a null set.

Sure event

This can be explained in 3 steps:
 1. Consider the experiment of rolling a die.
• We know that S = {1, 2, 3, 4, 5, 6}
2. Suppose that, we are interested in those outcomes which are either odd or even.
• Then we can pick out all six outcomes.
• We can write a set E using those outcomes:
E = {1, 2, 3, 4, 5, 6}
   ♦ We see that, E is same as S.
3. If E = S, then that event is a sure event.
• We can do the experiment any number of times we like. The outcome will always be an element of E. Because, all outcomes are present in E.

Simple event

This can be explained in 7 steps:
1. Consider the experiment of tossing a coin two times.
We know that S = {(H,H), (H,T), (T,H), (T,T)}
2. Suppose that, we are interested in those outcomes in which:
   ♦ First toss gives T.
   ♦ Second toss gives H.
• Then we can pick out only one outcome, which is (T,H).
• We can write a set E using this outcome:
E = {(T,H)}
3. If E has only one element, then that event is called a simple event.
4. In fact, we can pick out each element from S and write distinct sets.
   ♦ E₁ = {(H,H)}
   ♦ E₂ = {(H,T)}
   ♦ E₃ = {(T,T)}
• E₁ is the event in which both tosses give H.
• E₂ is the event in which first toss gives H and second toss gives T.
• E₃ is the event in which both tosses give T.
5. So we can write an important point:
If there are n elements in S, then there will be n simple events.
6. Consider the event in which there is atleast one H.
• This event is not a simple event because, the set of this event has more than one elements.
7. A simple event is also known as an elementary event.    

Compound event

This can be explained in 5 steps:
1. Consider the experiment of tossing a coin three times.
We know that S = {(H,H,H), (H,H,T), (H,T,H), (H,T,T), (T,H,H), (T,H,T), (T,T,H), (T,T,T)}
(see first example in exercise 16.1 of the previous section)
2. Suppose that, we are interested in those outcomes in which:
   ♦ Exactly one H is obtained.
• Then we can pick out three outcomes, which are (H,T,T), (T,H,T) and (T,T,H).
• We can write a set E using these outcomes:
E = {(H,T,T), (T,H,T), (T,T,H)}
3. If E has more than one element, then that event is called a compound event.
4. We can write more examples from this experiment. Let us write two such examples:
Example (i):
Suppose that, we are interested in those outcomes in which:
   ♦ Atleast one H is obtained.
• Then we can write:
   ♦ E₁ = {(H,H,H), (H,H,T), (H,T,H), (H,T,T), (T,H,H), (T,H,T), (T,T,H), (T,T,T)}
• E₁ is a compound event because there are more than one elements.
Example (ii):
Suppose that, we are interested in those outcomes in which:
   ♦ Atmost one H is obtained.
• Then we can write:
   ♦ E₂ = {(H,T,T), (T,H,T), (T,T,H), (T,T,T)}
• E₂ is a compound event because there are more than one elements.
5. So we can write an important point:
All subsets of S, which have more than one elements, are compound events.

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In the next section, we will see Algebra of Events.

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