Probability with Explanation

Q1. Three unbiased coins are tossed. What is the probability of getting at least 2 heads?

A. 1 3	B. 1 8
C. 1 2	D. 1 4

Answer- Option C

Explanation-

Here S = [TTT, TTH, THT, HTT, THH, HTH, HHT, HHH]
Let E = event of getting at least two heads = [THH, HTH, HHT, HHH].
P(E) =

n(E) n(S)

=

4 8

=

1 2

Q2. One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card?

A. 1 4	B. 1 13
C. 9 52	D. 4 13

Answer- Option D

Explanation-

Clearly, there are 52 cards, out of which there are 16 face cards.
P(getting a face card) =

16 52

=

4 13

Q3. One ticket is drawn at random from a bag containing 20 tickets numbered 1 to 20. What is the probability that the ticket was drawn having a number multiple of 3?

A. 3 10	B. 1 13
C. 1 2	D. 2 5

Answer- Option A

Explanation-

Here, S = {1, 2, 3, 4, ......, 19, 20}.
Let E = event of getting a multiple of 3.
i.e. = {3, 6, 9, 12, 15, 18}.
P(E) =

n(E) n(S)

=

6 20

=

3 10

.

Q4. From a pack of 52 cards, one card is drawn at random. What is the probability that the card drawn is ten or a spade?

A. 1 26	B. 1 13
C. 1 4	D. 4 13

Answer- Option D

Explanation-

Here, n(S) = 52.
There are 13 spades (including one ten) and there are 3 more tens.
Let E = event of getting a ten or a spade.
Then, n(E) = (13 + 3) = 16.
P(E) =

n(E) n(S)

=

16 52

=

4 13

.

Q5. The probability that a card is drawn from a pack of 52 cards will be a diamond or a king, is:

A. 4 13	B. 2 13
C. 1 13	D. 1 52

Answer- Option A

Explanation-

Here, n(S) = 52.
There are 13 cards of diamond (including one king) and there are 3 more kings.
Let E = event of getting a diamond or a king.
Then, n(E) = (13 + 3) = 16.
P(E) =

n(E) n(S)

=

16 52

=

4 13

.

Q6. A box contains 5 black, 4 orange and 3 white marbles. Three marbles are drawn at random. What is the probability that they are not of the same color?

A. 52 55	B. 3 55
C. 41 44	D. 3 44

Answer- Option C

Explanation-

Let S be the sample space. Then,
n(S) = number of ways of drawing 3 marbles out of 12
= ¹²C₃ =

(12 x 11 x 10) (3 x 2 x 1)

= 220

.

Let E be the event of drawing 3 balls of the same color.
Then, E = event of drawing (3 balls out of 5) or (balls out of 4) or (3 balls out of 3)
n(E) = (⁵C₃ + ⁴C₃ + ³C₃) = (⁵C₂ +⁴C₁ + 1)
=

(5 x 4) (2 x 1)

+ 4 + 1 = 15

.

P(E) =

n(E) n(S)

=

15 220

=

3 44

.

Required probability =

1 -

3 44

=

41 44

.

Q7. A bag contains 6 orange and 4 black balls. Three balls are drawn at random. What is the probability that one ball is black and the other two are orange?

A. 3 10	B. 1 2
C. 1 12	D. 3 10

Answer- Option B

Explanation-

Let S be the sample space. Then,
n(S) = number of ways of drawing 3 marbles out of 12
= ¹⁰C₃ =

(10 x 9 x 8) (3 x 2 x 1)

= 120

.

Let = event of drawing 1 black and 2 orange balls.
n(E) = Number of ways of drawing 1 black ball out of 4 and 2 orange balls out of 6.
n(E) = (⁴C₁ + ⁶C₂)
=

4 x

6 x 5 2 x 1

= 60.

P(E) =

n(E) n(S)

=

60 120

=

1 2

.