Probability
QUICK RECAP
CONDITIONAL PROBABILITY P (E ∩ F )
P (E /F ) = , where P(F) ≠ 0
P (F )
8 Let E and F be two events of a random X Properties of Conditional Probability
experiment, then the conditional probability (i) For an event A and sample space S,
of occurrence of E under the condition that F P(S/A) = P(A/A) = 1
has already occurred, i.e., P(E/F) is given by (ii) If A and B are any two events of a sample
, space S and F is an event of S such that THEOREM OF TOTAL PROBABILITY
P(F) ≠ 0, then
P((A∪B)/F)=P(A/F)+P(B/F)–P((A∩B)/F) 8 If E1, E2, E3, ..., En are mutually exclusive and
exhaustive events associated with a sample
In particular, if A and B are disjoint events,
space S of a random experiment and A is any
then P((A ∪ B)/F) = P(A/F) + P (B/F)
event associated with S, then
(iii) P(A′/B) = 1 – P(A/B)
P(A) = P(E1) P(A/E1) + P(E2) P(A/E2)
(iv) 0 < P(A/B) < 1
+ ... + P(En) P(A/En)
MULTIPLICATION THEOREM ON
BAYES’ THEOREM
PROBABILITY
8 If E1, E2, E3, ..., En are mutually exclusive and
8 Let A and B are two events associated with a exhaustive events associated with a random
sample space S. Then A ∩ B denotes the event
experiment and A is any event associated
that both A and B have occurred. The event
with the experiment, then
A ∩ B is also written as AB.
P (Ei )P ( A / Ei )
Also, P(A ∩ B) = P(A)⋅P(B/A) = P(B)⋅P(A/B), P (Ei /A) = ,
provided P(A) ≠ 0, P(B) ≠ 0 Σ P (Ei )P ( A / Ei )
i
Note : If A, B and C are three events associated where i = 1, 2, 3, ..., n
with a random experiment, then
P(A ∩ B ∩ C) = P(A)⋅P(B/A)⋅P(C/(A ∩ B)) RANDOM VARIABLE
= P(A)⋅P(B/A)⋅P(C/AB)
8 A random variable is a real valued function,
Similarly, multiplication rule of probability whose domain is the sample space of a
can be extended to four or more events. random experiment. Generally, it is denoted
INDEPENDENT EVENTS by X.
8 Events are said to be independent, if the PROBABILITY DISTRIBUTIONS
occurrence or non-occurrence of one does
8 Let real numbers x1, x2, ..., xn are the possible
not affect the probability of the occurrence values of a random variable X and p1, p2, ...,
or non-occurrence of the other. pn are the corresponding probabilities to
8 Two events A and B associated with a random each value of the random variable X. Then
experiment are said to be independent if the probability distribution is
(i) P(A/B) = P(A) provided P(B) ≠ 0 X : x1 x2 ... xn
(ii) P(B/A) = P(B) provided P(A) ≠ 0 P(X) : p1 p2 ... pn.
Note : Note :
(i) Two events A and B associated with a (i) 1 > pi > 0
random experiment are independent if (ii) Sum of probabilities (p1 + p2 + ... + pn) = 1
P(A ∩ B) = P(A) ⋅ P(B) X Mean : If X is a random variable, then mean
(ii) Two events A and B are said to be ( X ) of X is defined as
dependent if they are not independent, n
i.e., if P(A ∩ B) ≠ P(A) ⋅ P(B) X = m = p1x1 + p2x2 + ... + pnxn = ∑ pi xi
i =1
(iii) Events A1, A2, ..., An are mutually
independent if the probability of the Mean of a random variable is also called the
simultaneous occurrence of (any) finite expectation of X, denoted by E(X).
number of events is equal to the product X Variance : If X is a random variable, then
of their separate probabilities. While variance of X is defined as
2
these events are pairwise independent if n n
P(Ai ∩ Aj) = P(Ai)P(Aj) for all i ≠ j. Var ( X ) = σ = ∑
2
pi xi2 − ∑ pi xi
i =1 i =1
(iv) If A and B are independent events,
then so are the events A and B′; A′ and B; or Var(X) = E(X2) – {E(X)}2
A′ and B′. Note : Standard deviation (s) = Var( X )
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
13.1 Introduction 13.2 Conditional Probability
VSA (1 mark) VSA (1 mark)
1. Two cards are drawn at random and one- 7. A card is picked at random from a pack of
52 playing cards. Given that the picked card
by-one without replacement from a well-
is a queen, the probability of this card to be a
shuffled pack of 52 playing cards. Find the card of spade is
probability that one card is red and the other 1 4
is black. (2020) (a) (b)
3 13
2. A bag contains 3 black, 4 red and 2 green 1 1
(c) (d) (2020)
balls. If three balls are drawn simultaneously 4 2
at random, then the probability that the balls SA (2 marks)
are of different colours is . (2020)
8. 12 cards numbered 1 to 12 (one number on
3. A die is thrown once. Let A be the event that one card), are placed in a box and mixed up
the number obtained is greater than 3. Let B thoroughly. Then a card is drawn at random
be the event that the number obtained is less from the box. If it is known that the number
than 5. Then P(A ∪ B) is on the drawn card is greater than 5, find
the probability that the card bears an odd
2 3 number. (AI 2019)
(a) (b)
5 5
9. A black and a red die are rolled together.
(c) 0 (d) 1 (2020) Find the conditional probability of obtaining
the sum 8, given that the red die resulted in a
SA (2 marks)
number less than 4. (2018)
4. If A and B are two events such that P(A) = 0.4,
LA 1 (4 marks)
P(B) = 0.3 and P(A ∪ B) = 0.6, then find
P(B′ ∩ A). (2020) 10. Assume that each born child is equally likely
to be a boy or a girl. If a family has two
5. Out of 8 outstanding students of a school, in children, what is the conditional probability
which there are 3 boys and 5 girls, a team that both are girls? Given that
of 4 students is to be selected for a quiz (i) the youngest is a girl.
competition. Find the probability that 2 boys (ii) atleast one is a girl. (Delhi 2014)
and 2 girls are selected. (AI 2019) 11. A couple has 2 children. Find the probability
that both are boys, if it is known that
LA 1 (4 marks) (i) one of them is a boy,
(ii) the older child is a boy. (Delhi 2014C)
6. A bag A contains 4 black and 6 red balls and
bag B contains 7 black and 3 red balls. A die LA 2 (6 marks)
is thrown. If 1 or 2 appears on it, then bag
12. Consider the experiment of tossing a coin.
A is chosen, otherwise bag B. If two balls
If the coin shows head, toss it again, but if
are drawn at random (without replacement) it shows tail, then throw a die. Find the
from the selected bag, find the probability of conditional probability of the event that ‘the
one of them being red and another black. die shows a number greater than 4’ given
(Delhi 2015) that ‘there is at least one tail’. (Delhi 2014C)
QUICK RECAP
CONDITIONAL PROBABILITY P (E ∩ F )
P (E /F ) = , where P(F) ≠ 0
P (F )
8 Let E and F be two events of a random X Properties of Conditional Probability
experiment, then the conditional probability (i) For an event A and sample space S,
of occurrence of E under the condition that F P(S/A) = P(A/A) = 1
has already occurred, i.e., P(E/F) is given by (ii) If A and B are any two events of a sample
, space S and F is an event of S such that THEOREM OF TOTAL PROBABILITY
P(F) ≠ 0, then
P((A∪B)/F)=P(A/F)+P(B/F)–P((A∩B)/F) 8 If E1, E2, E3, ..., En are mutually exclusive and
exhaustive events associated with a sample
In particular, if A and B are disjoint events,
space S of a random experiment and A is any
then P((A ∪ B)/F) = P(A/F) + P (B/F)
event associated with S, then
(iii) P(A′/B) = 1 – P(A/B)
P(A) = P(E1) P(A/E1) + P(E2) P(A/E2)
(iv) 0 < P(A/B) < 1
+ ... + P(En) P(A/En)
MULTIPLICATION THEOREM ON
BAYES’ THEOREM
PROBABILITY
8 If E1, E2, E3, ..., En are mutually exclusive and
8 Let A and B are two events associated with a exhaustive events associated with a random
sample space S. Then A ∩ B denotes the event
experiment and A is any event associated
that both A and B have occurred. The event
with the experiment, then
A ∩ B is also written as AB.
P (Ei )P ( A / Ei )
Also, P(A ∩ B) = P(A)⋅P(B/A) = P(B)⋅P(A/B), P (Ei /A) = ,
provided P(A) ≠ 0, P(B) ≠ 0 Σ P (Ei )P ( A / Ei )
i
Note : If A, B and C are three events associated where i = 1, 2, 3, ..., n
with a random experiment, then
P(A ∩ B ∩ C) = P(A)⋅P(B/A)⋅P(C/(A ∩ B)) RANDOM VARIABLE
= P(A)⋅P(B/A)⋅P(C/AB)
8 A random variable is a real valued function,
Similarly, multiplication rule of probability whose domain is the sample space of a
can be extended to four or more events. random experiment. Generally, it is denoted
INDEPENDENT EVENTS by X.
8 Events are said to be independent, if the PROBABILITY DISTRIBUTIONS
occurrence or non-occurrence of one does
8 Let real numbers x1, x2, ..., xn are the possible
not affect the probability of the occurrence values of a random variable X and p1, p2, ...,
or non-occurrence of the other. pn are the corresponding probabilities to
8 Two events A and B associated with a random each value of the random variable X. Then
experiment are said to be independent if the probability distribution is
(i) P(A/B) = P(A) provided P(B) ≠ 0 X : x1 x2 ... xn
(ii) P(B/A) = P(B) provided P(A) ≠ 0 P(X) : p1 p2 ... pn.
Note : Note :
(i) Two events A and B associated with a (i) 1 > pi > 0
random experiment are independent if (ii) Sum of probabilities (p1 + p2 + ... + pn) = 1
P(A ∩ B) = P(A) ⋅ P(B) X Mean : If X is a random variable, then mean
(ii) Two events A and B are said to be ( X ) of X is defined as
dependent if they are not independent, n
i.e., if P(A ∩ B) ≠ P(A) ⋅ P(B) X = m = p1x1 + p2x2 + ... + pnxn = ∑ pi xi
i =1
(iii) Events A1, A2, ..., An are mutually
independent if the probability of the Mean of a random variable is also called the
simultaneous occurrence of (any) finite expectation of X, denoted by E(X).
number of events is equal to the product X Variance : If X is a random variable, then
of their separate probabilities. While variance of X is defined as
2
these events are pairwise independent if n n
P(Ai ∩ Aj) = P(Ai)P(Aj) for all i ≠ j. Var ( X ) = σ = ∑
2
pi xi2 − ∑ pi xi
i =1 i =1
(iv) If A and B are independent events,
then so are the events A and B′; A′ and B; or Var(X) = E(X2) – {E(X)}2
A′ and B′. Note : Standard deviation (s) = Var( X )
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
13.1 Introduction 13.2 Conditional Probability
VSA (1 mark) VSA (1 mark)
1. Two cards are drawn at random and one- 7. A card is picked at random from a pack of
52 playing cards. Given that the picked card
by-one without replacement from a well-
is a queen, the probability of this card to be a
shuffled pack of 52 playing cards. Find the card of spade is
probability that one card is red and the other 1 4
is black. (2020) (a) (b)
3 13
2. A bag contains 3 black, 4 red and 2 green 1 1
(c) (d) (2020)
balls. If three balls are drawn simultaneously 4 2
at random, then the probability that the balls SA (2 marks)
are of different colours is . (2020)
8. 12 cards numbered 1 to 12 (one number on
3. A die is thrown once. Let A be the event that one card), are placed in a box and mixed up
the number obtained is greater than 3. Let B thoroughly. Then a card is drawn at random
be the event that the number obtained is less from the box. If it is known that the number
than 5. Then P(A ∪ B) is on the drawn card is greater than 5, find
the probability that the card bears an odd
2 3 number. (AI 2019)
(a) (b)
5 5
9. A black and a red die are rolled together.
(c) 0 (d) 1 (2020) Find the conditional probability of obtaining
the sum 8, given that the red die resulted in a
SA (2 marks)
number less than 4. (2018)
4. If A and B are two events such that P(A) = 0.4,
LA 1 (4 marks)
P(B) = 0.3 and P(A ∪ B) = 0.6, then find
P(B′ ∩ A). (2020) 10. Assume that each born child is equally likely
to be a boy or a girl. If a family has two
5. Out of 8 outstanding students of a school, in children, what is the conditional probability
which there are 3 boys and 5 girls, a team that both are girls? Given that
of 4 students is to be selected for a quiz (i) the youngest is a girl.
competition. Find the probability that 2 boys (ii) atleast one is a girl. (Delhi 2014)
and 2 girls are selected. (AI 2019) 11. A couple has 2 children. Find the probability
that both are boys, if it is known that
LA 1 (4 marks) (i) one of them is a boy,
(ii) the older child is a boy. (Delhi 2014C)
6. A bag A contains 4 black and 6 red balls and
bag B contains 7 black and 3 red balls. A die LA 2 (6 marks)
is thrown. If 1 or 2 appears on it, then bag
12. Consider the experiment of tossing a coin.
A is chosen, otherwise bag B. If two balls
If the coin shows head, toss it again, but if
are drawn at random (without replacement) it shows tail, then throw a die. Find the
from the selected bag, find the probability of conditional probability of the event that ‘the
one of them being red and another black. die shows a number greater than 4’ given
(Delhi 2015) that ‘there is at least one tail’. (Delhi 2014C)
