18th july PROBABILITY - 2nd lecture continued..................
When it comes to real life scenario, probability is the term used everywhere whether it be in terms of household work educational work or buisness purpose. But to find out the exact value of probability in terms of numbers we deal with its mathematical interpretation
Probability is a
measure or estimation of how likely it is that something will happen or that a
statement is true. Probabilities are given a value between 0 (0% chance or will
not happen) and 1 (100%
chance or will happen). The higher the degree of probability,
the more likely the event is to happen, or, in a longer series of samples, the
greater the number of times such event is expected to happen.
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Probability consists of mainly 3 laws:
1) Additional law
2) Conditional law
3) Multiplication law
The Addition Law of Probability
If two events A
and B are mutually exclusive then
P (A∪B) = P(A)
+P(B)
This is the
simplified version of the Addition Law. However, when A and B are not mutually exclusive,
A∩B ≠ ∅, it can be shown that a more general law applies:
P(A ∪B) = P(A)
+P(B)−P(A ∩B)
Of course if A ∩
B = ∅ then, since P(∅) = 0 this general
expression reduces to the simpler version.
Lets take an
example to understand the above law:
A bag contains
20 marbles, 3 are colored red, 6 are colored green, 4 are colored blue, 2 are
colored white and 5 are colored yellow. One ball is selected at random. Find
the probabilities of the following events.
(a) the ball is
either red or green
(b) the ball is
not blue
(c) the ball is
either red or white or blue.
Answer
In the last
example (part (c)) we could alternatively have used an obvious extension of the
law
of addition for
mutually exclusive events:
P(R ∪W ∪B) = P(R)
+P(W) +P(B) = 3/20 + 2/20+ 4/20= 9/20.
The Conditional Law of Probability
We start with a random experiment that has a sample space S P . Suppose that we know that an event B A A A B B A B P(A∩B) .
However, conditional
probability, given that B 1/P(B) . Thus, we are led inexorably to the following
definition:
Let A and B be events in a random
experiment with P(B)>0 . The conditional probability of A given B is defined to be-
P(A|B)= P(A∩B) / P(B)
Suppose a bag contains 6 balls, 3 red and 3 white. Two balls are chosen
(without replacement)
at random, one after the other. Consider the two events A, B:
A is event “first ball chosen is red”
B is event “second ball chosen is red”
We easily find P (A) = 3/6=1/2
However, determining the probability of B is not quite so Straight forward.
if the first ball chosen is red then the
bag subsequently contains 2 red balls and 3 white. In this case P (B) = 2/5
However, if the first ball chosen is white then the bag subsequently
contains 3 red balls and 2 white. In this case P(B) = 3/5
What this example shows is that the probability that B occurs is
clearly dependent upon whether or not the event A has occurred. The probability
of B occurring is conditional on the occurrence or otherwise of A.
The conditional probability of an event B occurring given that event A
has occurred is written
P (B|A). In this particular example:
P(B|A) = 2/5 and P(B|A) = 3/5
The Multiplication Law of Probability
The probability of the intersection of two events
A and B is
P(A ∩ B) = P(A)P(B |A)=
P(B) P(A |B)
If A and B are independent, P(A ∩B) = P(A)P(B).
The Multiplicative Law of Probability is often
used to determine the probability of an event which involves a sequence of
random occurances.
Say that the probability that I am in my office at any given moment of the typical school day is .65 Also, say that the probability that someone is looking for me in my office at any given moment of the school day is .15 .What is the probability that that during some particular moment, I am in my office and someone looks for me there?
Written by- Poorva Saboo
Group Members-
Raghav Kabra
Abhishek Panwala
Pareena Neema
Poorva Saboo
Parita Mandhana
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