The document discusses key concepts in probability such as random experiments, outcomes, sample spaces, events, types of events including impossible, sure, simple, and compound events. It also covers algebra of events including unions, intersections, complements and mutually exclusive events. The document defines mutually exclusive and exhaustive events. Finally, it introduces the axiomatic approach to defining probability as a function that satisfies three axioms.
2. Random experiments
• In our day to day life, we perform many activities which have a fixed result
no matter any number of times they have been repeated. For example given
any triangle, without knowing the three angles, we can definitely say that
the sum of measure of angles is 180 degrees
• We also perform many experimental activities, where the result may not be
same when they are repeated under identical conditions. For example when
a coin is tossed it may turn up a head or a tail, but we are not sure which
one of these results will actually be obtained. Such experiments are called
random experiments
3. Random experiments
• An experiment is called a random experiment if it satisfies the following
two conditions:
• (i) It has more than one possible outcome
• (ii) It is not possible to predict the outcome in advance
5. Outcomes & sample spaces
• A possible result of a random experiment is
called its outcome.
• The set of all possible outcomes is called the
sample space of the experiment
6. Outcomes & sample space Example
• Consider the experiment of rolling a die.
• The outcomes of this experiment are 1, 2, 3, 4, 5 and 6
• The set of outcomes {1,2,3,4,5,6} is called the sample space of the
experiment
7. Outcomes & sample space Example
• In general, the set of all possible outcomes of a random experiment is
called the sample space associated with the experiment.
• Sample space is denoted by the letter S.
• Each element of the sample space is called a sample point.
• In other words, each outcome of the random experiment is also called a
sample point.
9. occurrence of an event
• Consider the experiment of throwing a die. Let E denotes the event “a
number less than 4 appears”.
• Hence the sample space S of the experiment is : S = { 1, 2, 3 }
• If actually ‘1’ had appeared on the die then we say that event E has
occurred. As a matter of fact if outcomes are 2 or 3, we say that event E
has occurred
• Thus, the event E of a sample space S is said to have occurred if the
outcome ω of the experiment is such that ω ∈ E.
• If the outcome ω is such that ω ∉ E, we say that the event E has not
occurred.
11. types of events
• Events can be classified into various types on the basis of the elements
they have:
1. Impossible and Sure Events: The empty set and the sample space S describe
an impossible event and a sure event respectively.
2. Simple Event: If an event E has only one element in the sample space S, it
is called a simple event.
3. Compound Event: If an event has more than one element in the sample
space S, it is called a compound event
13. Impossible and sure events
• The empty set and the sample space S are two subsets of S.
• The empty set is called an impossible event.
• The entire sample space S is called the sure event
14. Impossible events
• For an example, let us consider the rolling of a die. The associated sample
space is
S = {1, 2, 3, 4, 5, 6}
• Let E be the event “The number appears on the die is a multiple of 7”
• Clearly, no outcome satisfies the condition given in the event E. Thus the
empty set only corresponds to the event E.
• Therefore we say the event E is an impossible event
15. sure events
• Let’s take another event F: “The number turns up is odd or event”
• Therefore F = {1, 2, 3, 4, 5, 6}
• Thus the event F=S is called a sure event
17. simple and compound events
• Simple Event: If an event E has only one element in a sample space, it is
called a simple event
• Compound Event: If an event E has more than one element in a sample
space, it is called a compound event.
18. simple events
• For an example, let us consider the experiment of tossing two coins. Its
sample space is S = {HH, HT, TH, TT}
• There are four simple events: E1 = {HH}, E2={HT}, E3={TH}, E4={TT}
19. compound events
• Let’s take another event F: “The number turns up is odd or event”
• Therefore F = {1, 2, 3, 4, 5, 6}
• Thus the event F=S is called a sure event
21. Algebra of events
• In the chapter on sets we have studied about different ways of combining
two or more sets viz. union, intersection, difference and complement of a
set
• Like-wise we can combine two or more events by using the analogous set
notations.
22. Algebra of events
• Let A, B, C be events associated with an experiment whose sample space is
S.
1. Complementary Event For every event A, there corresponds another event A’
called the complementary event to A. It is also the called the event ‘not A’
2. The Event A or B The union of two events A and B contains all those
elements which are either in A or in B or both.
3. The Event ‘A and B’ The intersection of two events A and B is the events
consisting of those element which are common to both A and B
4. The Event ‘A but not B’ A-B is the event which consists of all those
elements which are in A but not in B.
24. Mutually Exclusive Events
• Two events A and B are called mutually exclusive events if the occurrence
of any one of them excludes the occurrence of the other event i.e. if they
can not occur simultaneously
25. Mutually Exclusive Events
• Consider the experiment of rolling a die, a sample space is S = {1, 2, 3, 4,
5, 6}.
• Let event A be ‘an odd number appears’. Let event B be ‘an even number
appears.
• Clearly event A excludes event B. In other words, the occurrence of A
excludes the occurrence of B and vice-versa.
• A = {1, 3, 5} and B = {2, 4, 6}
27. Exhaustive Events
• Consider an experiment of sample space S and three events A, B and C. If
A∪B∪C = S, then A, B, C are called exhaustive events.
• Let’s take the example of throwing a die. S = {1, 2, 3, 4, 5, 6}.
Event A = “a number less than 4 appears”
Event B = “a number greater than 2 but less than 5 appears”
Event C = “a number greater than 4 appears”
• Then A = {1, 2, 3}, B = {3, 4}, C = {5, 6}. Clearly A∪B∪C = S.
• Therefore A, B, C are called exhaustive events.
28. Mutually exclusive and Exhaustive Events
• In general, if E1, E2, E3, …, En are n events of a sample space S and if E1 ∪
E2 ∪ E3 ∪ … ∪ En = S, then E1, E2, …, En are called exhaustive events.
• Additionally, if Ei ∩ Ej = 𝛟 for i ≠ j i.e. events Ei and Ej are pairwise
disjoint
• Then events E1, E2, …, En are called mutually exclusive and exhaustive
events
30. Axiomatic approach
• Let S be the sample space of a random experiment.
• The probability P is a real valued function whose domain is the power set
of S and range is the interval [0,1] satisfying the following axioms:
i. For any event E, P (E) ≥ 0
ii. P(S) = 1
iii. If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F)
31. Axiomatic approach
• It follows from the axioms that P(𝛟) = 0 as can be seen below:
P(E ∪ 𝛟) = P(E) + P(𝛟) (Since E and 𝛟 are disjoint)
P(E) = P(E) + P(𝛟) => P(𝛟) = 0
It also follows that
• 0 ≤ P(wi) ≤ 1 for each wi ∈ S
• P(w1) + P(w2) + … + P(wn) = 1
• For any event A, P(A) = ΣP(wi), wi ∈ A
32. Mutually exclusive and Exhaustive Events
• In general, if E1, E2, E3, …, En are n events of a sample space S and if E1 ∪
E2 ∪ E3 ∪ … ∪ En = S, then E1, E2, …, En are called exhaustive events.
• Additionally, if Ei ∩ Ej = 𝛟 for i ≠ j i.e. events Ei and Ej are pairwise
disjoint
• Then events E1, E2, …, En are called mutually exclusive and exhaustive
events
33. Probability
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