Probability Test

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Probability Test - 43

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Weekly Quiz Competition

Question 1
1 / -0

Given two mutually exclusive events $$A$$ and $$B$$ such that $$P(A)=\dfrac 12$$ and $$P(B)=\dfrac 13$$, find $$P(A\ or\ B)$$

A
$$\dfrac{3}{5}$$

B
$$\dfrac{5}{6}$$

C
$$\dfrac{4}{5}$$

D
None of these

Solution

Given mutually exclusive events
$$\Rightarrow \ A\cap B=\phi$$
$$\therefore \ P(A\cup B)=P(A)+P(B)-P(A\cap B)$$
$$\Rightarrow \ P(A\cup B)=\dfrac {1}{2}+\dfrac {1}{3}-0$$
$$\Rightarrow \ P(A\cup B)=\dfrac {3+2}{6}$$
$$\Rightarrow \ P(A\ or\ B)=P(A\cup B)=\dfrac {5}{6}$$
Question 2
1 / -0

Choose the most appropriate option.
$$12$$ defective pens are accidentally mixed with $$132$$ good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.

A
$$7/12$$

B
$$11/12$$

C
$$10/12$$

D
$$9/12$$

Solution

Total number of pens $$=$$ Total defective pens $$+$$ Total good pens
$$=12+132$$
$$=144$$
Number of good pens $$=132$$
Probability of getting good pen $$=\dfrac{Number\,of\,good\,pens}{Total\,number\, of\, pens}$$

$$=\dfrac{132}{144}$$

$$=\dfrac{11}{12}$$
Question 3
1 / -0

A card is picked at random from a pack of $$52$$ playing cards. given that the picked card is a queen, the probability of this card to be a card of spade is

A
$$\dfrac{1}{3}$$

B
$$\dfrac{4}{13}$$

C
$$\dfrac{1}{4}$$

D
$$\dfrac{1}{2}$$

Solution

No. of spade cards $$= 13$$

No. of queen in spade set $$=1$$

$$\therefore n$$(spade queen) $$= 13_{c_1}$$

$$\therefore n$$ (Total no. of cases) $$= 52_{c_1}$$

$$\therefore P$$ (spade queen) $$= \dfrac{13_{c_1}}{52_{c_1}}$$

$$= \dfrac{13}{52} = \dfrac{1}{4}$$

option C
Question 4
1 / -0

A quadratic equation $$ax^2 + bx + c = 0$$, with distinct coefficients is formed. It a, b, c are chosen from the numbers $$2, 3, 5$$ then the probability that the equation has real roots is

A
$$\dfrac{1}{3}$$

B
$$\dfrac{2}{5}$$

C
$$\dfrac{1}{4}$$

D
$$\dfrac{1}{5}$$

E
$$\dfrac{2}{3}$$

Solution

Total number of ways of assigning values $$2,3,5$$ to $$a,b,c,=3!=6$$
Now, for quadratic equation $$ax^2+bx+c=0$$ to have real roots $$b^2-4ac \ge 0$$. This is possible only when $$a=2,b=5,c=3$$ or $$a=3,b=5,c=2$$
$$\Rightarrow$$ Required probability $$=\dfrac{2}{6}=\dfrac{1}{3}$$
Question 5
1 / -0

Two dice of different colours are thrown at a time. The probability that the sum is either $$7$$ or $$11$$ is

A
$$\dfrac{7}{36}$$

B
$$\dfrac{2}{9}$$

C
$$\dfrac{2}{3}$$

D
$$\dfrac{5}{9}$$

E
$$\dfrac{6}{7}$$

Solution

event $$A=$$ sum is either $$7$$ or $$11$$

Probability of sum of $$7$$
$$=(6,1),(5,2),(4,3),(3,4),(2,5),(1,6)$$ and probability of sum of $$11$$
$$=(6,5),(5,6)$$
$$P(A)=\dfrac{6}{36}+\dfrac{2}{36}$$
$$=\dfrac{8}{36}=\dfrac{2}{9}$$
Question 6
1 / -0

A coin is tossed $$5$$ times. What is the probability that tail appears an odd number of times?

A
$$\cfrac{3}{5}$$

B
$$\cfrac{2}{15}$$

C
$$\cfrac{1}{2}$$

D
$$\cfrac{1}{3}$$

Solution
Question 7
1 / -0

If $$A$$ and$$B$$ are mutually exclusive events such that $$P(A)=0.4,P(B)=x$$ and $$P(A\cup B)=0.5$$, then $$x=$$?

A
$$0.2$$

B
$$0.1$$

C
$$\cfrac{4}{5}$$

D
none of these

Solution
Question 8
1 / -0

In a class, $$60$$% of the students read mathematics, $$25$$% biology and $$15$$% both mathematics and biology. One student is selected at random. What is the probability that he reads mathematics, if it is known that he reads biology?

A
$$\cfrac{2}{5}$$

B
$$\cfrac{3}{5}$$

C
$$\cfrac{3}{8}$$

D
$$\cfrac{5}{8}$$

Solution
Question 9
1 / -0

A fair coin is tossed $$6$$ times. What is the probability of getting at least $$3$$ heads?

A
$$\cfrac{11}{16}$$

B
$$\cfrac{21}{32}$$

C
$$\cfrac{1}{18}$$

D
$$\cfrac{3}{64}$$

Solution
Question 10
1 / -0

A die is thrown twice and the sum of the numbers appearing is observed to be $$7$$. What is the conditional probability that the number $$2$$ has appeared at least once?

A
$$\cfrac{1}{6}$$

B
$$\cfrac{1}{3}$$

C
$$\cfrac{2}{7}$$

D
$$\cfrac{3}{5}$$

Solution

$${\textbf{Step -1: Find required events.}}$$

$${\text{Let S be the sample space of a die is thrown twice.}}$$

$$\Rightarrow n(S) = 36.$$

$${\text{Let X be the event getting the sum is 7 and Y be the event getting that 2 appears at least once.}}$$

$$\Rightarrow P(X)= \dfrac{6}{36}$$

$$\Rightarrow P(Y)= \dfrac{11}{36}$$

$${\text{So,}}$$ $$X \cap Y =$$ $${\text{{(2, 5), (5, 2)}}}$$

$${\textbf{Step -2: Find required probability.}}$$

$$\Rightarrow P(X \cap Y) = \dfrac{2}{36}$$

$${\text{Required probability(P(Y|X))}}$$ $$= \dfrac{P(X \cap Y)}{P(X)}= \dfrac{2}{6}= \dfrac{1}{3}.$$

$${\textbf{Hence, the option B is correct.}}$$

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Probability Test - 43

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Probability Test - 43

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Question 1

$$\dfrac{3}{5}$$

$$\dfrac{5}{6}$$

$$\dfrac{4}{5}$$

None of these

Solution

Question 2

$$7/12$$

$$11/12$$

$$10/12$$

$$9/12$$

Solution

Question 3

$$\dfrac{1}{3}$$

$$\dfrac{4}{13}$$

$$\dfrac{1}{4}$$

$$\dfrac{1}{2}$$

Solution

Question 4

$$\dfrac{1}{3}$$

$$\dfrac{2}{5}$$

$$\dfrac{1}{4}$$

$$\dfrac{1}{5}$$

$$\dfrac{2}{3}$$

Solution

Question 5

$$\dfrac{7}{36}$$

$$\dfrac{2}{9}$$

$$\dfrac{2}{3}$$

$$\dfrac{5}{9}$$

$$\dfrac{6}{7}$$

Solution

Question 6

$$\cfrac{3}{5}$$

$$\cfrac{2}{15}$$

$$\cfrac{1}{2}$$

$$\cfrac{1}{3}$$

Solution

Question 7

$$0.2$$

$$0.1$$

$$\cfrac{4}{5}$$

none of these

Solution

Question 8

$$\cfrac{2}{5}$$

$$\cfrac{3}{5}$$

$$\cfrac{3}{8}$$

$$\cfrac{5}{8}$$

Solution

Question 9

$$\cfrac{11}{16}$$

$$\cfrac{21}{32}$$

$$\cfrac{1}{18}$$

$$\cfrac{3}{64}$$

Solution

Question 10

$$\cfrac{1}{6}$$

$$\cfrac{1}{3}$$

$$\cfrac{2}{7}$$

$$\cfrac{3}{5}$$

Solution

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