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11. A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is:

A. $\frac{1}{13}$

B. $\frac{2}{13}$

C. $\frac{1}{26}$

D. $\frac{1}{52}$

Discuss

Solution:

Here, n(S) = 52
Let E = event of getting a queen of club or a king of heart
Then, n(E) = 2

\begin{aligned} ∴ P (E) = \frac{n (E)}{n (S)} \\ = \frac{2}{52} \\ = \frac{1}{26} \end{aligned}

12. A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is:

A. $\frac{1}{22}$

B. $\frac{3}{22}$

C. $\frac{2}{91}$

D. $\frac{2}{77}$

Discuss

Solution:

Let S be the sample space
Then, n(S) = number of ways of drawing 3 balls out of 15

\begin{aligned} =^{15} C_{3} \\ = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} \\ = 455 \end{aligned}

Let E = event of getting all the 3 red balls

\begin{aligned} ∴ n (E) =^{5} C_{3} =^{5} C_{2} \\ = \frac{5 \times 4}{2 \times 1} = 10 \\ ∴ P (E) = \frac{n (E)}{n (S)} \\ = \frac{10}{455} \\ = \frac{2}{91} \end{aligned}

13. Two cards are drawn together from a pack of 52 cards. The probability that one is a spade and one is a heart, is:

A. $\frac{3}{20}$

B. $\frac{29}{34}$

C. $\frac{47}{100}$

D. $\frac{13}{102}$

Discuss

Solution:

Let S be the sample space

\begin{aligned} n (S) =^{52} C_{2} = \frac{52 \times 51}{2 \times 1} \\ = 1326 \end{aligned}

Let E = event of getting 1 spade and 1 heart
∴ n(E) = number of ways of choosing 1 spade out of 13 and 1 heart out of 13

\begin{aligned} =^{13} C_{1} \times^{13} C_{1} \\ = 13 \times 13 \\ = 169 \\ ∴ P (E) = \frac{n (E)}{n (S)} = \frac{169}{1326} = \frac{13}{102} \end{aligned}

14. One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King only)?

A. $\frac{1}{13}$

B. $\frac{3}{13}$

C. $\frac{1}{4}$

D. $\frac{9}{52}$

Discuss

Solution:

Clearly, there are 52 cards, out of which there are 12 face cards.

\begin{aligned} ∴ P (getting a face card) \\ = \frac{12}{52} = \frac{3}{13} \end{aligned}

15. A bag contains 6 black and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?

A. $\frac{3}{4}$

B. $\frac{4}{7}$

C. $\frac{1}{8}$

D. $\frac{3}{7}$

Discuss

Solution:

\begin{aligned} Let number of balls \\ = (6 + 8) \\ = 14 \\ Number of white balls = 8 \\ P (drawing a white ball) \\ = \frac{8}{14} = \frac{4}{7} \end{aligned}

16. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn bears a number which is a multiple of 3 ?

A. $\frac{3}{10}$

B. $\frac{3}{20}$

C. $\frac{2}{5}$

D. $\frac{1}{2}$

Discuss

Solution:

Here, S = {1, 2, 3, 4,........, 19, 20}
Let E = even of getting a multiple of 3 = {3, 6, 9, 12, 15, 18}

∴ P (E) = \frac{n (E)}{n (S)} = \frac{6}{20} = \frac{3}{10}

17. From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the cards being kings?

A. $\frac{1}{15}$

B. $\frac{25}{57}$

C. $\frac{35}{256}$

D. $\frac{1}{221}$

Discuss

Solution:

Let S be the sample space
Then,

n (S) =

^{52} C_{2} =

\frac{(52 \times 51)}{(2 \times 1)}

=1326
Let E = event of getting 2 kings out of 4
∴

n (E) =^{4} C_{2} =

\frac{(4 \times 3)}{(2 \times 1)}

= 6
∴

P (E) = \frac{n (E)}{n (S)} =

\frac{6}{1326} = \frac{1}{221}

18. An urn contains 6 red, 4 blue, 2 green and 3 yellow marbles. If 4 marbles are picked up at random, what is the probability that at least one of them is blue?

A. $\frac{4}{15}$

B. $\frac{69}{91}$

C. $\frac{11}{15}$

D. $\frac{22}{91}$

Discuss

Solution:

Total number of marbles = (6 + 4 + 2 + 3) = 15
Let E be the event of drawing 4 marbles such that none is blue.
Then, n(E) = number of ways of drawing 4 marbles out of 11 non-blue.

^{11} C_{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1}

= 330
And n(S) =

^{15} C_{4} =

\frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1}

= 1365

∴ P (E) = \frac{n (E)}{n (S)}

= \frac{330}{1365}

= \frac{22}{91}

∴ Required probality =

(1 - \frac{22}{91})

= \frac{69}{91}

19. In a simultaneous throw of two coins, the probability of getting at least one head is-

A. $\frac{1}{2}$

B. $\frac{1}{3}$

C. $\frac{2}{3}$

D. $\frac{3}{4}$

Discuss

Solution:

Here S = {HH, HT, TH, TT}
Let E = event of getting at least one head = {HT, TH, HH}

∴ P (E) = \frac{n (E)}{n (S)} = \frac{3}{4}

20. A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is-

A. $\frac{1}{13}$

B. $\frac{2}{13}$

C. $\frac{1}{26}$

D. $\frac{1}{52}$

Discuss

Solution:

Hence, n(S) = 52
Let E = event of getting a queen of club or a king of heart.
Then, n(E) = 2

∴ P (E) = \frac{n (E)}{n (S)} = \frac{2}{52} = \frac{1}{26}

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