2. Definitions of various term
1) Sample space : The set of all possible outcomes of a trial (random experiment) is
called its sample space.
• It is generally denoted by S and each outcome of the trial is said to be a sample
point.
2) Event : An event is a subset of a sample space
(i) Simple event : An event containing only a single sample point is called an
elementary or simple event.
(ii) Compound events : Events obtained by combining together two or more
elementary events are known as the compound events or decomposable events.
(iii) Mutually exclusive or disjoint events : Events are said to be mutually exclusive or
disjoint or incompatible if the occurrence of any one of them prevents the occurrence
of all the others.
3. Definitions of various term
(iv) Mutually non-exclusive events : The events which are not mutually exclusive are known as
compatible events or mutually non exclusive events.
(v) Independent events : Events are said to be independent if the happening (or non-happening) of
one event is not affected by the happening (or non- happening) of others.
(vi) Dependent events : Two or more events are said to be dependent if the happening of one
event affects (partially or totally) other event.
4. BASIC PROBABILITY
EQUATION:
P=Number of favourable outcomes
Total Number of Outcomes
P(Getting one red ball)= Number of Red balls
Total number of balls
= 4
10
P+NP=1
Total probability=1
6. QUESTIONS
1)What will be the possibility of drawing a jack or a spade from a well shuffled
standard deck of 52 playing cards?
7. QUESTIONS
2)A box has 6 black,4 red,2 white and 3 blue shirts. When 2 shirts are picked randomly, What is the probability
that either both are white or both are blue?
8. QUESTIONS
3)A pot has 2 white,6 black,4 grey and 8 green balls. If one ball is picked randomly from the pot,
what is the probability of it being black or green?
9. QUESTIONS
4)In a set of 30 game cards,17 are white and rest are green. 4 white and 5 green are marked IMPORTANT.If a
card is choosen randomly from this set,what is the probability of choosing a green card or an ‘IMPORTANT’
card?
10. QUESTIONS
5) There are 2 pots.One pot has 5 red and 3 green marbles.Other has 4 red and 2 green marbles.What is
the probability of drawing a red marble?
11. QUESTIONS
6)On rolling a dice 2 times,the sum of 2 numbers that appear on the uppermost face is 8.What is the
probability that the first throw of dice yields 4?
12. CONDITIONAL PROBABILITY
Conditional probability is the probability of an event occurring given that another
event has already occurred. It's calculated as the probability of both events
happening divided by the probability of the condition event. Symbolically, it's
expressed as P(A|B) = P(A ∩ B) / P(B), where P(A|B) is the probability of event A
given event B has occurred, P(A ∩ B) is the probability of both events A and B
happening, and P(B) is the probability of event B occurring. It's used to assess the
likelihood of an event in light of some additional information.
13. CONDITIONAL PROBABILITY
Let A and B be two events associated with a random experiment.
Then, the probability of occurrence of A under the condition that B has already occurred
and P(B) 0, is called the conditional probability and it is denoted by P(A/B).
Thus, P(A/B) = Probability of occurrence of A, given that B has already happened.
=P(A ∩ B)
P(B)
Similarly, P(B/A) = Probability of occurrence of B, given that A has already happened.
=P(A ∩ B)
P(A)
14. PROPERTIES
Property 1: P(S|F) = P(F|F) = 1
Property 2: If A and B are any two events of a sample space S and F is an event of S
such that P(F) ≠ 0, then P((A B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)
∪
Property 3: P(E |F) = 1 − P(E|F)
′
Example: If P(A) = 7 13 , P(B) = 9 13 and P(A ∩ B) = 4 13 , evaluate P(A|B).
Solution: We have
15. QUESTIONS
1) A family has two children. What is the probability that both the children are boys
given that at least one of them is a boy ?
16. QUESTIONS
2) Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it
is known that the number on the drawn card is more than 3, what is the probability that it is an even number?
17. QUESTIONS
3) A die is thrown three times. Events A and B are defined as below:
A : 4 on the third throw
B : 6 on the first and 5 on the second throw
Find the probability of A given that B has already occurred.
18. MUTUALLY EXCLUSIVE EVENTS
Mutually exclusive events in probability are events that cannot occur simultaneously.
If one event happens, the other cannot.
For example, when flipping a coin, getting heads and getting tails are mutually
exclusive events. They cannot happen at the same time.
In probability theory, two events are said to be mutually exclusive if they cannot occur at the same
time or simultaneously. In other words, mutually exclusive events are called disjoint events. If
two events are considered disjoint events, then the probability of both events occurring at the
same time will be zero.
If A and B are the two events, then the probability of disjoint of event A and B is written by:
Probability of Disjoint (or) Mutually Exclusive Event = P ( A and B) = 0
If A and B are said to be mutually exclusive events then the probability of an event A occurring or
the probability of event B occurring that is P (a b) formula is given by P(A) + P(B), i.e.,
∪
P (A B) = P(A) + P(B)
∪
19. QUESTIONS
Question 1: What is the probability of a die showing a number 3 or
number 5?
Solution: Let,
P(3) is the probability of getting a number 3
P(5) is the probability of getting a number 5
P(3) = 1/6 and P(5) = 1/6
So,
P(3 or 5) = P(3) + P(5)
P(3 or 5) = (1/6) + (1/6) = 2/6
P(3 or 5) = 1/3
Therefore, the probability of a die showing 3 or 5 is 1/3.
20. QUESTIONS
2) A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that the
card drawn is a king or an ace.
Solution:
As per the definition of mutually exclusive events, selecting an ace and selecting a king from a well-
shuffled deck of 52 cards are termed mutually exclusive events.
Assume X to be the event of drawing a king and Y to be the event of drawing an ace.
In a standard deck of 52 cards, there exists 4 kings and 4 aces.
P (an event) = count of favorable outcomes / total count of outcomes
P (selecting a king from a standard deck of 52 cards) = P (X) = 4 / 52 = 1 / 13
P (selecting an ace from a standard deck of 52 cards) = P (Y) = 4 / 52 = 1 / 13
To compute P (king or ace).
By the formula of addition theorem for mutually exclusive events,
P (X U Y) = P (X) + P (Y)
P (X U Y) = (1 / 13) + (1 / 13)
= (1 + 1) / 13
= 2 / 13
The probability of selecting a king or an ace from a well-shuffled deck of 52 cards = 2 / 13.
21. INDEPENDENT EVENTS
Independent events in probability are events where the occurrence of one
event does not affect the occurrence of the other. Mathematically, two events A
and B are independent if the probability of both events happening is equal to
the product of their individual probabilities. In other words, P(A ∩ B) = P(A) *
P(B). For example, when flipping a coin and rolling a die, the outcome of one
event (coin flip) does not influence the outcome of the other event (die roll),
making them independent events.
24. BAYES THEOREM
Bayes theorem (also known as the Bayes Rule or Bayes Law) is used to determine
the conditional probability of event A when event B has already occurred.
The general statement of Bayes’ theorem is “The conditional probability of an
event A, given the occurrence of another event B, is equal to the product of the
event of B, given A and the probability of A divided by the probability of event
B.” i.e.
P(A|B) = P(B|A)P(A) / P(B)
