Probability Test 13

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Probability Test 13

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Weekly Quiz Competition

Question 1
1 / -0

We draw two cards from a deck of shuffled cards without replacement. Find the probability of getting the second card a queen.

A
$$\dfrac{1}{13}$$

B
$$\dfrac{2}{13}$$

C
$$\dfrac{5}{13}$$

D
None of these

Solution

There are two cases here:
Case 1: First card chosen is a queen
$$\dfrac 4{52}\times \dfrac 3{51}=\dfrac {1}{221}$$
Case 2: First card chosen is not a queen.
$$\dfrac {48}{52}\times \dfrac 4{51}=\dfrac {16}{221}$$
Adding both the cases, we get the probability of getting the second card a queen. $$\dfrac {17}{221} =\dfrac 4{52} = \dfrac 1{13}$$
Question 2
1 / -0

Tossing a coin is an example of .........

A
Infinite discrete sample space

B
Finite sample space

C
Continuous sample space

D
None of these

Solution

A coin is tossed.
The possible outcomes are "$$\text{head,tail}$$"
Thus $$ S=\{H,T\}$$
We get a finite sample space.
Thus tossing a coin is an example of finite sample space.
Question 3
1 / -0

For a random experiment, all possible outcomes are called

A
numerical space.

B
event space.

C
sample space.

D
both b and c.

Solution

We know that,
An outcome is a result of a random experiment.
The set of all possible outcomes is called the sample space.
The subset of the sample space is called event space.
Thus, for a random experiment, all possible outcomes are called sample space and event space.
Thus option $$(D)$$ is correct.
Question 4
1 / -0

A man is know to speak the truth $$3$$ out if $$4$$ times. He throws a die and reports that it is a six. The probability that it is actually a six is:

A
$$\dfrac{3}{8}$$

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

C
$$\dfrac{3}{4}$$

D
None of these
Question 5
1 / -0

Sample space for experiment in which a dice is rolled is

A
$$4$$

B
$$8$$

C
$$12$$

D
None of these

Solution

A dice is rolled.
Thus $$S=\{1,2,3,4,5,6\}$$
$$\Rightarrow n(S)=6$$.
That is, sample space for experiment in which a dice is rolled is $$6$$.
Since $$6$$ is not listed in the given options we choose D.
Question 6
1 / -0

Let A, B, C be three events such that A and B are independent and $$P(C) = 0$$, then events A, B, C are

A
independent

B
pairwise independent but not totally independent

C
$$P(A) = P(B) = P(C)$$

D
none of these

Solution

Given three events $$A,B,C.$$
A and B are independent $$\xi$$ $$P(C)=0.$$
A term 'independent event' means probability of events. They have a common outcomes and each event is non-empty.
As A and B are independent
i.e, $$P(A)\neq 0$$ and $$P(B)\neq 0.$$
For $$P(C)$$ to be independent,
$$\Rightarrow P(A\cap B)=P(A).P(B)$$ $$P(A\cap B\cap C)=P(A).P(B).P(C)$$
$$\Rightarrow P(A\cap C)=P(A).P(C)$$
$$\Rightarrow P(B\cap C)=P(B).P(C)$$
If all the given events are true, then A,B,C are independent.
Hence, the answer is independent.
Question 7
1 / -0

Sample space for experiment in which two coins are tossed is

A
$$8$$

B
$$4$$

C
$$2$$

D
None of these

Solution

Two coins are tossed.
Number of outcomes in sample space when two coins are tossed is given by $$2^2=4$$.
Hence, sample space for experiment in which two coins are tossed is $$4$$.
Question 8
1 / -0

Choosing a birthdate is an example of .........

A
Infinite discrete sample space

B
Finite sample space

C
Continuous sample space

D
None of these

Solution

Let us see the sample space for choosing a birthdate.
A person can be born in any date of a month and a month has maximum of $$30$$ or $$31$$ days.
Therefore, $$S=\{1,2,3,4,5,6,7,8,9,10,11,12,13,...,30,31\}$$ which is a finite set.
Thus choosing a birthdate is an example of finite sample space.
Question 9
1 / -0

One bag contains 3 white balls, 7 red balls and 15 black balls. Another bag contains 10 white balls, 6 red balls and 9 black balls. One ball is taken from each bag. What is the probability that both the balls will be of the same colour?

A
$$207/625$$

B
$$191/625$$

C
$$23/625$$

D
$$227/625$$

Solution

$$\textbf{Step-1: Find the number of balls in each bag.}$$

$$\text{Bag}$$ $$I$$ $$\text{ contains 3 white, 7 red and 15 black balls.}$$

$$\text{Bag}$$ $$II$$ $$\text{contains 10white, 6 red and 9 black balls.}$$

$$\therefore$$ $$\text{Each bag contains total of}$$ $$ 25$$ $$\text{balls.}$$

$$\textbf{Step-2: Find the number of cases for the selection of a ball.}$$

$$\text{There are three cases for selection of a particular ball:-}$$

$$\text{Case 1:}$$ $$\text{Probability of getting one white ball each from Bag}$$ $$I$$ $$\text{and Bag}$$ $$II=\dfrac{3}{25}\times\dfrac{10}{25}$$

$$\text{Case 2:}$$ $$\text{Probability of getting one red ball each from Bag}$$ $$I$$ $$\text{and Bag}$$ $$II=\dfrac{7}{25}\times\dfrac{6}{25}$$

$$\text{Case 3:}$$ $$\text{Probability of getting one black ball each from Bag}$$ $$I$$ $$\text{and Bag}$$ $$II=\dfrac{15}{25}\times\dfrac{9}{25}$$

$$\textbf{Step-3: Add the above cases to get the required probability.}$$

$$\therefore$$ $$\text{Total probability that both the ball will be of same colour}$$ $$=\dfrac{30} {625}+\dfrac{42}{625}+\dfrac{135}{25}=\dfrac{207}{625}.$$

$$\textbf{Hence, the answer is}$$ $$\boldsymbol{\dfrac{207}{625}}.$$
Question 10
1 / -0

An experiment can result in only $$3$$ mutually exclusive events $$A, B$$ and $$C$$. If $$P(A) = 2P(B) = 3P(C)$$, then $$P(A) =$$

A
$$\dfrac{6}{11}$$

B
$$\dfrac{5}{11}$$

C
$$\dfrac{9}{11}$$

D
None

Solution

Given that $$P(A)=2P(B)=3P(C)$$
Let $$P(C)=x$$ , we get $$P(B)=\cfrac{3}{2}x$$ and $$P(A)=3x$$
Given that there are only these three events.
So we have $$P(A)+P(B)+P(C)=1$$
$$\Rightarrow 3x+\cfrac{3x}{2}+x=1$$
$$\Rightarrow x=\cfrac{2}{11}$$
Therefore the value of $$P(A)=3x=\cfrac{6}{11}$$
So the correct option is $$A$$.

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Probability Test 13

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Probability Test 13

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SHARING IS CARING

Question 1

$$\dfrac{1}{13}$$

$$\dfrac{2}{13}$$

$$\dfrac{5}{13}$$

None of these

Solution

Question 2

Infinite discrete sample space

Finite sample space

Continuous sample space

None of these

Solution

Question 3

numerical space.

event space.

sample space.

both b and c.

Solution

Question 4

$$\dfrac{3}{8}$$

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

$$\dfrac{3}{4}$$

None of these

Question 5

$$4$$

$$8$$

$$12$$

None of these

Solution

Question 6

independent

pairwise independent but not totally independent

$$P(A) = P(B) = P(C)$$

none of these

Solution

Question 7

$$8$$

$$4$$

$$2$$

None of these

Solution

Question 8

Infinite discrete sample space

Finite sample space

Continuous sample space

None of these

Solution

Question 9

$$207/625$$

$$191/625$$

$$23/625$$

$$227/625$$

Solution

Question 10

$$\dfrac{6}{11}$$

$$\dfrac{5}{11}$$

$$\dfrac{9}{11}$$

None

Solution

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