Algebra II: Chapter 12 Probability and Statistics Section ...
12.probability
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Probability1. Introduction
2. Definitions of various Terms
3. Mathematical or Classical Probability,
4. Statistical or Empirical Probability,
5. Mathematical Tools,
6. Sets and Elements of sets,
7. Operations on sets,
8. axiomatic Approach to Probability,
9. Probability function,
10. Mathematical Law of Addition of Probabilities
11. Conditional Probabilities,
12. Independent Events,
13. Bayes Theorem,
14. Geometric Probability. 1
Introduction 1. The theory of probability was developed towards the end of the 18th
century and its history suggests that it developed with the study of games
and chance, such as rolling a dice, drawing a card, flipping a coin etc.
2. ‘probability’ thus suggest that there is an uncertainty about the happening
of events
3. Probability is the chance that something will happen - how likely it is
that some event will happen
4. Sometimes you can measure a probability with a number: "10% chance of
rain", or you can use words such as impossible, unlikely, possible, even
chance, likely and certain.
2
Probability- Basic Term
Trial:-A procedure or an experiment to
collect any statistical data such as rolling
a dice or flipping a coin is called a trial.
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Probability- Basic Term
Random Trial or Random Experiment:- When the
outcome of any experiment can not be predicted
precisely then the experiment is called a random trial
or random experiment
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Probability- Basic Term
Sample Space:- The totality of all the outcomes or
results of a random experiment is denoted by Greek
alphabet W or English alphabets and is called the
sample space
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Probability- Basic Term
Event:- Any subset of a sample space is
called an event
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Probability- Basic Term
Equally Likely Events:- All possible results of a
random experiment are called equally likely
outcomes and we have no reason to expect any one
rather than the other.
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Probability- Basic Term
Mutually Exclusive Events :- Events are called
mutually exclusive or disjoint or incompatible if the
occurrence of one of them precludes the occurrence
of all the others.
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Probability- Basic Term
Exhaustive Events:- Events are exhaustive
when they include all the possibilities
associated with the same trial
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Probability Independent Events:-Two events are said to be independent
if the occurrence of any event does not affect the occurrence
of the other event.
Dependent Events:- If the occurrence or non-occurrence of
any event affects the happening of the other, then the events
are said to be dependent events
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Definitions of Probability
We shall now consider two definitions of
probability :
(1) Mathematical or a priori or classical.
(2) Statistical or empirical.
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Mathematical or a priori or classical Definitions of Probability
1. If there are ‘n’ exhaustive, mutually exclusive and equally
likely cases and m of them are favorable to an event A,
The probability of A happening is defined as the ratio m/n
Expressed as a formula :-
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2. This definition is due to ‘Laplace.’ Thus probability is a
concept which measures numerically the degree of certainty or
uncertainty of the occurrence of an event.
Example Probability
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Example Probability
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Van Mise’s Statistical (or Empirical) Definitions of Probability
1. The classical definition of probability has a disadvantage i.e. the words
‘equally likely’ are vague.
2. In fact, since these words seem to be synonymous with "equally probable".
This definition is circular as it is defining (in terms) of itself.
3. Therefore, the estimated or empirical probability of an event is taken as the
relative frequency of the occurrence of the event when the number of
observations is very large.
4. If trials are to be repeated a great number of times under essentially
the same condition then the limit of the ratio of the number of times
that an event happens to the total number of trials, as the number of
trials increases indefinitely is called the probability of the happening
of the event.
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Van Mise’s Statistical (or Empirical) Definitions of Probability
1. If trials are to be repeated a great number of times under
essentially the same condition then the limit of the ratio of the
number of times that an event happens to the total number of
trials, as the number of trials increases indefinitely is called
the probability of the happening of the event.
2. The two definitions are apparently different but both of them
can be reconciled the same sense.
3. Symbolically p (A) = p =
4. provided it is finite and unique.
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Example Probability
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Example1 Find the probability of getting heads in tossing a coin.
Solution : Experiment : Tossing a coin
Sample space : S = { H, T} n (S) = 2
Event A : getting heads
A = { H}
n (A) = 1
Therefore, p (A) =
Example Probability
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Example2 Find the probability of getting 3 or 5 in throwing a die
Solution :
Experiment : Throwing a dice
Sample space : S = {1, 2, 3, 4, 5, 6 } Þ n (S) = 2
Event A : getting 3 or 6
A = {3, 6} n (A) = 2
Therefore, p (A) =
Example Probability
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Example3 Two dice are rolled. Find the probability that the score on the
second die is greater than the score on the first die.
Solution :
Experiment : Two dice are rolled
Sample space : S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2),
(2, 3), (2, 4), (2, 6)}...
(6, 1), (6, 2) (, 3), (6, 4), (6, 5), (6, 6) }
n (S) = 6 ´ 6 = 36
Event A : The score on the second die > the score on the 1st die.
i.e. A = { (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 3), (2, 4), (2, 5), (2, 6) (3, 4),
(3, 5), (3, 6) (4, 5), (4, 6) (5, 6)}
n (A) = 15
Therefore, p (A) =
Example Probability
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Example4 A ball is drawn at random from a box containing 6 red balls, 4 whiteballs and 5 blue balls. Determine the probability that the ball drawn is
red
Solution :
Let R denote the events of drawing a red ball.