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Probability



Q1. An urn contains 3 pink and 6 white balls . A, B, C and D draw one ball in turn without replacement, one who gets pink ball first wins . find the probability of winning of C.

Solution.
If A and B both will lose then C will get chance to win
probability of winning of C = P(A' ∩ B' ∩ C)
= P(A') . P(B') . P(C) =
6/9
.
5/8
.
3/7
=
5/28
Solution

Q2. A card is drawn at random from a pack of 52 cards . find the probability of getting a club or a king .

In this card n(S) = 52

P(A or B) = P(A ∪ B)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

A = shows king , B = shows club

P(A ∪ B) =
4/52
+
13/52
-
1/52
=
4/13


Direct one can do this , there are 13 club cards and 4 king but one king is already exist in 13 club cards so favourable outcome are 16 , probability is
16/52
=
4/13
Solution

Q3. Two dice are thrown simultaneously, find the probability of getting the sum as a prime number.

(a) 5/12 (b) 1/12 (c) 1/2 (d) 5/14

Solution.
(a) 5/12
Solution

Q4. A bag contains cards which are numbered from 6 to 80. A card is drawn at random from the bag. Find the probability that it has 2 digit number.

(a) 6/80 (b) 71/76 (c) 71/75 (d) 63/100

Solution.
n(S) = 75

n(A) = 71

P(A) = n(A)/n(S)

= 71/75
Solution

Q5. 3 books of maths , 2 books of physics and 4 books of statistics are randomly arranged on a shelf in a vertical row . find the probability that the book of the same language are together .

Solution.
Total number of ways to arrange the book are 9! and
the number of ways to arrange the books of same language together = 3! . 2! . 4! . 3!
the required probability is
3! . 2! . 4! . 3!/9!
Solution

Q6. A, B and C are three mutually exclusive and exhaustive events such that P(A) = 2P(B) = 4P(C) find the value of P(C) .

Solution.
A, B and C are exhaustive events

so P(A) + P(B) + P(C) = 1

let P(A) = 2P(B) = 4P(C) = x

P(A) = x , P(B) =
x/2
, P(C) =
x/4


x +
x/2
+
x/4
= 1 ⇒
3x/2
+
x/4
= 1

14x/8
= 1 ⇒
7x/4
= 1 ⇒ x =
4/7


P(C) =
x/4
=
4/7.4
=
1/7
Solution

Q7. Probability of occurrance of an event + Probability of non-occurrance of that event is

(a) 1 (b) 0 (c) 2 (d) none of these

Solution.
(a) 1
Solution

Q8. A boy contains 5 Red balls, 4 blue balls and x balls . If the random probability of picking two green ballsis
1/7
, find x .

Solution.
1/7
=
xC2/9 + xC2
=
x!/2!(x - 2)!
/
(9 + x)!/(7 + x)! 2!


1/7
=
x!/(x - 2)!
x
(7 + x)!/(9 + x)!




1/7
=
x (x - 1)(x - 2)!/(x - 2)!
x
(7 + x)!/(9 + x)(8 + x)(7 + x)!


1/7
=
x(x - 1)/(9 + x)(8 + x)

solving this x = 6
Solution

Q9. An unbiased coin is tossed three times. The expected value of the number of heads is .

(a) 1 (b) 1.5 (c) 2.5 (d) 3

Solution.
(b) 1.5
Solution

Q10. Conditional probability P(A/B) is defined only when
(a) A is an sure event
(b) B is an certain event .
(c) B is not an impossible event .
(d) None of these .

Solution.
(c) B is not an impossible event .
P(A/B) =
P(A ∩ B)/P(B)

∴ P(B) ≠ 0
Solution