Maths notes probability

PROBABILITY
MAIN CONCEPTS AND RESULTS
** Random Experiments : An experiment is called 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.
** Outcomes and sample space : A possible result of a random experiment is called its outcome.
The set of all possible outcomes of a random experiment is called the sample space associated
with the experiment. Each element of the sample space is called a sample point. Any subset E of a
sample space S is called an event.
** Impossible and Sure Events : The empty set φ and the sample space S describe events. φ is called
an
impossible event and S, i.e., the whole sample space is called the sure event.
** Compound Event : If an event has more than one sample point, it is called a Compound event.
** Complementary Event : For every event A, there corresponds another event A  called the
complementary event to A. It is also called the event „not A‟.
** The Event ‘A or B’ : When the sets A and B are two events associated with a sample space, then
„A  B‟ is the event „either A or B or both‟. This event „A  B‟ is also called „A or B‟.
** The Event ‘A and B’ : If A and B are two events, then the set A ∩ B denotes the event „A and B‟.
** The Event ‘A but not B’ : the set A – B denotes the event „A but not B‟. A – B = A ∩ B´
** 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. In
this case the sets A and B are disjoint i.e. A ∩ B = φ.
** Exhaustive events : if E1, E2, ..., En are n events of a sample space S and if
n
E1  E2  E3  …  En =  E i = S , then E1, E2, ...., En are called exhaustive events.
i 1
n
if Ei ∩ Ej = φ for i ≠ j i.e., events Ei and Ej are pairwise disjoint and  E i = S , then events E1, E2,
i 1

..., En are
called mutually exclusive and exhaustive events.
** Axiomatic Approach to Probability : 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).
From the axiomatic definition of probability it follows that
(i) 0 ≤ P (ωi) ≤ 1 for each ωi  S
(ii) P (ω1) + P (ω2) + ... + P (ωn) = 1
(iii) For any event A, P(A) = Σ P(ωi ), ωi  A.
** Equally likely outcomes : All outcomes with equal probability.
** Probability of an event: For a finite sample space with equally likely outcomes


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, n A 
Probability of an event P(A) = (A) , where n(A) = number of elements in the set A,
n S
n(S) = number of elements in the set S.

** Probability of the event ‘A or B’ : P(A or B) = P(A  B) = P(A) + P(B) − P(A∩B).
For mutually exclusive events A and B, we have P(A  B) P(A) P(B)
** Probability of event ‘not A’ = P( A′ ) = P(not A) = 1 – P(A).

Example 1:A box contains 1 red and 3 identical white balls. Two balls are drawn at random in
succession without replacement. Write the sample space for this experiment.

Solution:Let R denotes the red ball and W denotes the white ball.
Given that a box contains 1 red and 3 identical white ball.
To draw two balls at random in succession without replacement, the sample space can be written as:
S = {RW, WR, WW}= {(R,W), (W,R),(W,W)}
Example2: An experiment involves rolling a pair of dice and recording the numbers that come up.
Describe the following events:
A: the sum is greater than 8.
B: 2 occurs on either die
C: the sum is at least 7 and a multiple of 3.
Which pairs of these events are mutually exclusive?
Solution:Given that a pair of dice 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, 5), (2, 6)
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3),
(5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
n(S) = 36
Event A: The sum is greater than 8
A = {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)}
Event B: 2 occurs on either die
B = {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)}
Event C: The sum is at least 7 and a multiple of 3
C = {(3, 6), (4, 5), (5, 4), (6, 3), (6, 6)}
Here,
A∩B=Φ
B∩C=Φ
A∩C≠Φ
Therefore, the pair of events A, B and B, C are mutually exclusive.
Examples3: What is the probability of getting the number 6 at least once in a regular die if it can
roll it 6 times?
Let A be the event that 6 does not occur at all.
Now, the probability of at least one 6 occurs = 1 – P(A)
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