Probability 1-2

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7Two unbiased dice are thrown simultaneously. What is the probability that the sum of the numbers is 10?
A1/12
B1/14
C1/15
D1/16

Answer: A

Explanation:
Total number of possible outcomes $=\; 6^2 = 36$
Number of outcomes that the sum of the numbers is 10 = 3 ($\because$ 5+5, 4+6, 6+4)
$\therefore$ Required probability $ = \dfrac{3}{{36}} = \dfrac{1}{{12}}$

8A bag contains 4 blue balls and 5 green balls. A ball is drawn at random. What is the probability that it is blue in colour?
A2/9
B4/9
C5/9
D6/9

Answer: B

Explanation:
Total number number of ball $= 4 + 5 = 9$
Total number of ways of drawing one ball $ = {}^9{C_1} = 9$
Total number of ways of drawing one blue ball $ = {}^4{C_1} = 4$
$\therefore$ Required probability $ = \dfrac{4}{{9}}$

9A bag contain 4 green marbles and 5 yellow marbles. Two marbles are drawn at random from the bag. What is the probability that both the balls are of same colour?
A3/9
B5/9
C4/9
D6/9

Answer: C

Explanation:
Total number number of balls $= 4 + 5 = 9$
Number of ways in which the two marbles drawn are of same colour $ = {}^4{C_2} + {}^5{C_2}$$ = \dfrac{{4 \times 3}}{{2 \times 1}} + \dfrac{{5 \times 4}}{{2 \times 1}}$$ = 16$
Number of ways of drawing $2$ marbles out of $9$ marbles $ = {}^9{C_2} = \dfrac{{9 \times 8}}{{2 \times 1}} = 36$
$\therefore$Required probability $ = \dfrac{{16}}{{36}} = \dfrac{4}{9}$

10If a card is drawn at random from a well shuffled pack of cards, find the probability of drawing a red card?
A5/8
B2/8
C1/2
D4/6

Answer: C

Explanation:
Number of ways of selecting one card from a pack of cards $ = {}^{52}{C_1} = 52$
$\spadesuit$ = 13 Spades
$\color{red}{\heartsuit}$ = 13 Hearts
$\color{red}{\diamondsuit}$ = 13 Diamonds
$\clubsuit$ = 13 Clubs
Total red cards $= 13 + 13 = 26$
$\therefore$ Required probability $ = \dfrac{{26}}{{52}} = \dfrac{1}{2}$

11If a card is drawn at random from a well shuffled pack of cards, find the probability of drawing an honour?
A2/13
B3/13
C7/13
D4/13

Answer: D

Explanation:
$\spadesuit$ = 13 Spades; $\;\color{red}{\heartsuit}$ = 13 Hearts; $\;\color{red}{\diamondsuit}$ = 13 Diamonds; $\;\clubsuit$ = 13 Clubs
Honour $=A,\;J,\;Q,\;K$
Each suite contains 4 honour cards mentioned above.
Hence there are a total of 16 honour cards in a pack.
$\therefore$ Required probability $ = \dfrac{{16}}{{52}} = \dfrac{4}{{13}}$

12Two numbers are selected at random from the first 25 natural numbers. What is the probability that the sum of two numbers is 14?
A1/50
B2/25
C2/50
D3/50

Answer: A

Explanation:
Number of outcomes that the sum of two numbers is 14 is $(1,13),(2,12)$,$(3,11),(4,10)$,$(5,9),(6,8)$ $= 6$
Number of ways of selecting two numbers out of 25 $ = {}^{25}{C_2} = \dfrac{{25 \times 24}}{{2 \times 1}} = 300$
$\therefore$ Required probability $ = \dfrac{6}{{300}} = \dfrac{1}{{50}}$

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