Probability Test

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Probability Test - 22

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Weekly Quiz Competition

Question 1
1 / -0

Three unbiased coins are tossed, what is probability of getting exactly two heads ?

A
$$\displaystyle \frac{1}{3}$$

B
$$\displaystyle \frac{3}{4}$$

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

D
$$\displaystyle \frac{3}{8}$$

Solution

Possible outcomes of tossing three coins are:
$$ (HHH), (HHT), (HTH), (THH), (TTT), (TTH), (THT),(HTT) $$
here $$H $$ and $$ T$$ are denoted for Head and Tail.
Total no. of outcomes $$= 8$$
no. of outcomes with exactly two heads $$=$$$$ 3$$
$$\therefore $$ required probability = $$ \dfrac 38 $$
$$\therefore $$ Option D is correct.
Question 2
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Two die are thrown. Find the probability of the event that the sum of the numbers on their upper faces is $$7$$:

A
$$\displaystyle \frac{1}{6}$$

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

C
$$\displaystyle \frac{1}{9}$$

D
$$\displaystyle \frac{2}{7}$$

Solution

Total number of outcomes $$= 36$$
Favourable outcomes $$(6,1), (1,6), (2,5), (5,2), (4,3), (3,4) = 6$$
Probability $$=\dfrac{6}{36} = \dfrac{1}{6}$$
Question 3
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Two die are thrown. Find the probability of the event that the product of numbers on their upper faces is $$12$$:

A
$$\displaystyle \frac{1}{12}$$

B
$$\displaystyle \frac{1}{10}$$

C
$$\displaystyle \frac{1}{9}$$

D
$$\displaystyle \frac{1}{6}$$

Solution

Total number of outcomes $$= 36$$
Favourable outcomes $$(4,3), (3,4), (6,2), (2,6) = 4$$
Probability $$=\dfrac{4}{36} = \dfrac{1}{9}$$
Question 4
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All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting black face card

A
$$\displaystyle \frac{6}{49}$$

B
$$\displaystyle \frac{3}{49}$$

C
$$\displaystyle \frac{5}{49}$$

D
$$\displaystyle \frac{4}{49}$$

Solution

Total number of possibilities = $$49$$ (Since, 3 cards of spade are removed)
Number of black face cards = $$3$$ (3 cards of clubs)
Thus, required probability = $$\dfrac{3}{49}$$
Question 5
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A die is thrown twice. What is the probability that
$$(i)$$ 3 will not come up either time?
$$(ii)$$ 6 will come up at least once?

A
$$(i) \displaystyle\frac{12}{25} \\ (ii) \displaystyle\frac{16}{25}$$

B
$$(i) \displaystyle\frac{18}{25} \\ (ii)\displaystyle\frac{9}{25}$$

C
$$(i) \displaystyle\frac{25}{36} \\ (ii) \displaystyle\frac{11}{36}$$

D
$$(i) \displaystyle\frac{23}{36} \\ (ii) \displaystyle\frac{17}{36}$$

Solution

When a dice is thrown twice, the possible 36 outcomes.
$$ 3 $$ will not come up either time, when pair of outcomes are anything except $$ (1,3), (3, 1), (3, 3), (2, 3), (3,2), (3,6), (6,3) (4,3), (3,4) , (3,5),(5,3) $$

Probability that $$ 3 $$ will not come up either time $$ = \dfrac {36-11}{36} = \dfrac {25}{36} $$

$$ 6 $$ will come up at least once, when pair of outcomes are $$ (1,6), (6,1), (6,2), (2,6), (3,6), (6,3), (4,6), (6,4), (6,5), (5,6), (6,6) $$

Probability that $$ 6 $$ will come up at least once $$ = \dfrac {11}{36} $$
Question 6
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What is the probability that a number selected from the numbers 1, 2, 3, 4, 5......,16 is a prime number ?

A
$$\displaystyle \frac{1}{16}$$

B
$$\displaystyle \frac{5}{8}$$

C
$$\displaystyle \frac{3}{8}$$

D
$$\displaystyle \frac{7}{16}$$

Solution

Since, Total outcomes $$= 16$$
prime no's from $$1$$ to $$16$$ are: $$ 2,3,5,7,11,13 $$
total prime no's $$= 6$$
$$\therefore $$ favorable outcomes$$= 6$$
$$\therefore $$ probability = $$\dfrac {6}{16} = \dfrac 38 $$
$$\therefore $$ Option C is correct.
Question 7
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What is the probability that a leap year has $$53$$ Sundays?

A
$$\displaystyle\frac{1}{7}$$

B
$$\displaystyle\frac{12}{52}$$

C
$$\displaystyle\frac{2}{7}$$

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

Solution

A leap year has $$366$$ days = $$ 52 \times 7 + 2 $$ days
That means there will be $$52$$ Sunday/Monday.../Saturdays plus $$2$$ additional days. Now these $$2 $$ additional days would be combination of any of the successive two days.
i)Sunday $$+$$ Monday
ii)Monday $$+$$ Tuesday
iii)Tuesday $$+$$ Wednesday
iv)Wednesday $$+$$ Thursday
v)Thursday $$+$$ Friday
vi)Friday $$+$$ Saturday
vii)Saturday $$+$$ Sunday
Now the probability of two Sundays will be (i) & (vii) $$ = 2$$ outcomes out of above $$7$$ outcomes. So the probability of having two additional Sundays
$$= 2/7.$$
Question 8
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Three unbiased coins are tossed. What is the probability of getting at most 2 tails ?

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

B
$$\displaystyle \frac{5}{8}$$

C
$$\displaystyle \frac{7}{8}$$

D
$$\displaystyle \frac{1}{4}$$

Solution

Possible outcomes of tossing three coins are:
$$(HHH), (HHT), (HTH), (THH), (TTT), (TTH), (THT),(HTT)$$
where, $$H$$ and $$T$$ are represent "heads" and "tail".
Total no. of outcomes $$= 8$$
No. of outcomes with at the most two tails $$= 7$$
$$\therefore $$ Required probability $$= $$$$ \dfrac 78 $$
$$\therefore $$ Option C is correct.
Question 9
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A card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a spade or a king ?

A
$$\displaystyle \frac{4}{13}$$

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

C
$$\displaystyle \frac{2}{13}$$

D
$$\displaystyle \frac{1}{13}$$

Solution

Since, Total cards = $$52 $$
No. of kings = $$ 4 $$
No. of cards that are spade = $$ 13$$
There is one card of king which is of spade.
$$\therefore $$ no. of cards which are spade or a king = $$ 16$$
$$\therefore $$ Required probability = $$\displaystyle \frac {16}{52} = \frac 4{13} $$
$$\therefore $$ Option A is correct.
Question 10
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If a ball is drawn from a bag containing 20 balls of different colours, then probability of a white ball is _____.

A
$$\dfrac{1}{10}$$

B
$$\dfrac{1}{15}$$

C
$$\dfrac{1}{20}$$

D
$$\dfrac{1}{25}$$

Solution

There are 20 different coloured balls in the bag, 1 of them is white,
p(white)=$$\cfrac{1}{20}$$

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Re-Attempt Weekly Quiz Competition

Probability Test - 22

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Probability Test - 22

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SHARING IS CARING

Question 1

$$\displaystyle \frac{1}{3}$$

$$\displaystyle \frac{3}{4}$$

$$\displaystyle \frac{2}{3}$$

$$\displaystyle \frac{3}{8}$$

Solution

Question 2

$$\displaystyle \frac{1}{6}$$

$$\displaystyle \frac{1}{3}$$

$$\displaystyle \frac{1}{9}$$

$$\displaystyle \frac{2}{7}$$

Solution

Question 3

$$\displaystyle \frac{1}{12}$$

$$\displaystyle \frac{1}{10}$$

$$\displaystyle \frac{1}{9}$$

$$\displaystyle \frac{1}{6}$$

Solution

Question 4

$$\displaystyle \frac{6}{49}$$

$$\displaystyle \frac{3}{49}$$

$$\displaystyle \frac{5}{49}$$

$$\displaystyle \frac{4}{49}$$

Solution

Question 5

$$(i) \displaystyle\frac{12}{25} \\ (ii) \displaystyle\frac{16}{25}$$

$$(i) \displaystyle\frac{18}{25} \\ (ii)\displaystyle\frac{9}{25}$$

$$(i) \displaystyle\frac{25}{36} \\ (ii) \displaystyle\frac{11}{36}$$

$$(i) \displaystyle\frac{23}{36} \\ (ii) \displaystyle\frac{17}{36}$$

Solution

Question 6

$$\displaystyle \frac{1}{16}$$

$$\displaystyle \frac{5}{8}$$

$$\displaystyle \frac{3}{8}$$

$$\displaystyle \frac{7}{16}$$

Solution

Question 7

$$\displaystyle\frac{1}{7}$$

$$\displaystyle\frac{12}{52}$$

$$\displaystyle\frac{2}{7}$$

$$\displaystyle\frac{1}{52}$$

Solution

Question 8

$$\displaystyle \frac{1}{2}$$

$$\displaystyle \frac{5}{8}$$

$$\displaystyle \frac{7}{8}$$

$$\displaystyle \frac{1}{4}$$

Solution

Question 9

$$\displaystyle \frac{4}{13}$$

$$\displaystyle \frac{3}{13}$$

$$\displaystyle \frac{2}{13}$$

$$\displaystyle \frac{1}{13}$$

Solution

Question 10

$$\dfrac{1}{10}$$

$$\dfrac{1}{15}$$

$$\dfrac{1}{20}$$

$$\dfrac{1}{25}$$

Solution

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