Probability Test

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Probability Test - 59

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Question 1
1 / -0

Let $${ U }_{ 1 }$$ and $${ U }_{ 2 }$$ be two urns such that $${ U }_{ 1 }$$ contains 3 white and 2 red balls, and $${ U }_{ 2 }$$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $${ U }_{ 1 }$$ and put into $${ U }_{ 2 }.$$ However, if tail appears then 2 balls are drawn at random from $${ U }_{ 1 }$$ and put into $${ U }_{ 2 }.$$ Now 1 ball is drawn at random from $${ U }_{ 2 }.$$
The probability of the drawn ball from $${ U }_{ 2 }$$being white is-

A
$$\dfrac { 13 }{ 30 } $$

B
$$\dfrac { 23 }{ 30 } $$

C
$$\dfrac { 19 }{ 30 } $$

D
$$\dfrac { 1 }{ 30 } $$

Solution
Question 2
1 / -0

Three coins are thrown simultaneously 60 times, with the following frequencies:
No. of heads 3 2 1 0
Frequency 10 5 18 27
Based on these information find the probability of

A
P (getting 3 heads)

B
P (getting no heads)

C
P (at most 1 head)

D
P (at least 1 head)
Question 3
1 / -0

The events $$A$$ and $$B$$ are mutually exclusive, so

A
Prob ( $$A$$ or $$B$$ ) $$=$$ Prob $$( A ) +$$ Prob $$( B )$$

B
Prob ( $$A$$ and $$B$$ ) $$=$$ Prob $$( A ) \times$$ Prob $$( B )$$

C
Prob $$( A ) =$$ Prob $$( B )$$

D
Prob $$( A ) +$$ Prob $$( B ) = 1$$

Solution
Question 4
1 / -0

The slope of given line is

A
$$-4$$

B
$$-\dfrac { 1 }{ 4 } $$

C
$$-1$$

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

Solution
Question 5
1 / -0

An unbiased coin is tossed n times. Let X denote the number of times head occurs. If P(X=4), P(X=5) and P(X=6) are in AP, then the value of n can be

A
9

B
10

C
12

D
14

Solution
Question 6
1 / -0

The p.m.f.of a r.v. X is as follows ;
$$P(X=0)=3{ K }^{ 3 }\quad ,\quad P(X=1)=4K-10{ K }^{ 2 },\quad P(X=2)=5K-1,\quad P(X-x)=0$$ for any other value of x. then k is equal to

A
1

B
2

C
$$\frac { 1 }{ 3 } $$

D
$$\frac { 2 }{ 3 } $$

Solution
Question 7
1 / -0

If a number is selected at random, then getting a prime or composite number is a ________ event.

A
Mutually exclusive

B
Equally likely

C
$$0$$

D
None of the above

Solution

Two events are said to be mutually exclusive events, when both the events cannot occur simultaneously.

A number cannot be a prime number and composite number both. It is either a prime number or a composite number. Hence, the events are mutually exclusive.
Question 8
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Ravi and Rashmi are each holding 2 red cards and 2 black cards (all four red and all four black cards are identical). Ravi picks a card at random from Rashmi, and then Rashmi picks a card at random from Ravi. This process is repeated a second time. Let p be the probability that both have all 4 cards of the same colour. Then p satisfies

A
$$p \le 5\%$$

B
$$5\% < p \le 10\%$$

C
$$10\% < p \le 15\%$$

D
$$15\% < p$$

Solution

If both have all $$4$$ cards of the same color, then there are two possibilities at the end.
Possibility $$1$$ : Ravi holds $$4$$ red cards and Rashmi $$4$$ black cards.
Probability of this possibility $$=$$P('Ravi picks red in first pick' And 'Rashmi picks black in first pick' AND 'Ravi picks red in second pick' AND 'Rashmi pick black in second pick')
All $$4$$ picks are independent
$$=\dfrac{2}{4}\times\dfrac{2}{5}\times\dfrac{1}{4}\times\dfrac{1}{5}=\dfrac{1}{100}$$
Posibility $$2$$ : Ravi holds $$4$$ black cards and Rashmi $$4$$ red cards similarly probability $$=\dfrac{1}{100}$$
$$\therefore\,p=\dfrac{1}{50}=0.02=2\%$$
Question 9
1 / -0

A fair die is tossed repeatedly until a 6 is obtained. Let X denote the number of tosses required.
The probability that $$x \geq 3$$ equals

A
125/216

B
25/36

C
5/36

D
25/216

Solution
Question 10
1 / -0

Two cards are drawn from a well shuffled deck of 52 playing cards with replacement.
The probability, that both cards are queen, is

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

B
$$\dfrac{1}{13} + \dfrac{1}{13}$$

C
$$\dfrac{1}{13} \times \dfrac{1}{17}$$

D
$$\dfrac{1}{13} \times \dfrac{4}{51}$$

Solution