Probability Test

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Probability Test - 43

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Weekly Quiz Competition

Question 1
1 / -0

If the events A and B are independent if $$P(A')=\cfrac{2}{3}$$ and $$P(B')=\cfrac{2}{7}$$, then $$P(A \cap B)$$ is equal to.

A
$$\dfrac{5}{21}$$

B
$$\dfrac{3}{21}$$

C
$$\dfrac{4}{21}$$

D
$$\dfrac{1}{21}$$

Solution

$$P(A)=1-P(A')=1-\dfrac{2}{3}=\dfrac{1}{3}$$

and $$P(B)=1-P(B')=1-\dfrac{2}{7}=\dfrac{5}{7}$$

Now if A and B are independent event then,

$$P(A\cap B)=P(A)P(B)=\dfrac{1}{3}\cdot \dfrac{5}{7}=\dfrac{5}{21}$$
Question 2
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 $$A$$.
Question 3
1 / -0

Tickets numbered $$1$$ to $$20$$ are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of $$3$$ or $$5$$?

A
$$\dfrac{7}{20}$$

B
$$\dfrac{8}{20}$$

C
$$\dfrac{6}{20}$$

D
$$\dfrac{9}{20}$$

Solution

$$\textbf{Step -1: Finding the number of elements in the required event .}$$

$$\text{Let}$$ $$A$$ $$\text{be the event where the ticket drawn has a number which is a multiple of}$$ $$3/5.$$

$$\text{Hence,}$$ $$n(A)=9$$ $$\text{as the event will contain elements}$$ $$3,6,9,12,15,18,5,10,20.$$

$$\textbf{Step -2: Finding the probability}$$

$$\text{We know that}$$ $$P(A)=\dfrac{n(A)}{n(S)}$$

$$\text{Here,}$$ $$n(S)=20$$

$$P(A)=\dfrac{9}{20}$$

$$\textbf{Thus, the probability that the ticket drawn has a number which is a multiple of}$$ $$\mathbf{3}$$ $$\textbf{or}$$ $$\mathbf{5}$$ $$\textbf{is}$$ $$\mathbf{\dfrac{9}{20}.}$$
Question 4
1 / -0

One card is drawn at random from a pack of $$52$$ cards. What is the probability that the card drawn is a face card (Jack, Queen and King only)?

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

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

C
$$\dfrac{3}{13}$$

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

Solution

Clearly, there are $$52$$ cards, out of which there are $$12$$ face cards.
$$P (getting\ a\ face\ card) =\dfrac {\text{Number of face cards}}{\text{Total number of cards}}=\dfrac {12}{52}=\dfrac {3}{13}$$
Question 5
1 / -0

If $$P(A) + P(B) = 1$$; then which of the following option explains the event $$A$$ and $$B$$ correctly?

A
Event $$A$$ and $$B$$ are mutually exclusive, exhaustive and complementary events

B
Event $$A$$ and $$B$$ are mutually exclusive and exhaustive events

C
Event $$A$$ and $$B$$ are mutually exclusive and complementary events

D
Event $$A$$ and $$B$$ are exhaustive and complementary events

Solution

Mutually exclusive events are those events which cannot happen at the same time.
Mutually exhaustive events are those events in which atleast one of the events should occur.
Atleast one of $$A,B$$ should happen , both $$P(A),P(B)$$ cannot be zero at same time because given that $$P(A)+P(B)=1$$
Complementary events means those two events are the only possible events.
Here only $$A$$ and $$B$$ are possible events because $$P(A)+P(B)=1$$
Therefore Event $$A$$ and $$B$$ are mutually exclusive , exhaustive and complementary events.
So the correct option is $$A$$.
Question 6
1 / -0

Probability of sure event is

A
$$1$$

B
$$0$$

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

D
$$2$$

Solution

Probability of sure event is always $$1$$
Question 7
1 / -0

Which one can represent a probability of an event

A
$$\dfrac {7}{4}$$

B
$$-1$$

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

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

Solution

$$P < 1$$

Optoion (D) is correct $$2/3$$
Question 8
1 / -0

Directions For Questions

Choose the correct answer:

...view full instructions

If A and B are two events such that $$P(B)=0.25, P(B)=0.05$$ and $$P(A\cap B)=0.14$$, then $$P(A \cup B)=$$

A
0.61

B
0.16

C
0.14

D
0.6

Solution
Question 9
1 / -0

Two similar boxes $$B_{i}(i = 1, 2)$$ contains $$(i + 1)$$ red and $$(5 - i - 1)$$ black balls. One box is chosen at random and two balls are drawn randomly. What is the probability that both the balls are of different colours?

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

B
$$\dfrac{3}{10}$$

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

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

Solution

Clearly, $$B_1$$ has 2 red and 3 black balls whereas $$B_2$$ has 3 red and 2 black balls.
The probability of choosing a box randomly is $$\dfrac{1}{2}$$.
Assume that $$B_1$$ is chosen first and that red ball is drawn first and black ball second. Then, the probability of this event is $$\dfrac{1}{2}\times\dfrac{2}{5}\times\dfrac{3}{4}=\dfrac{3}{20}$$
Now, assume that $$B_1$$ is chosen first and that black ball is drawn first and red ball second. Then, the probability of this event is $$\dfrac{1}{2}\times\dfrac{3}{5}\times\dfrac{2}{4}=\dfrac{3}{20}$$
Now, assume that $$B_2$$ is chosen first and that black ball is drawn first and red ball second. Then, the probability of this event is $$\dfrac{1}{2}\times\dfrac{2}{5}\times\dfrac{3}{4}=\dfrac{3}{20}$$
Now, assume that $$B_2$$ is chosen first and that red ball is drawn first and black ball second. Then, the probability of this event is $$\dfrac{1}{2}\times\dfrac{3}{5}\times\dfrac{2}{4}=\dfrac{3}{20}$$
Hence, probability that one box is picked at random and the outcome of draw is 2 balls of different color is the sum of all the above described events=$$4\times\dfrac{3}{20}=\dfrac{3}{5}$$.
This is the required solution.
Question 10
1 / -0

Probability of impossible event is

A
$$1$$

B
$$0$$

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

D
$$-1$$

Solution

(P) Of possible event is $$0$$

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Re-Attempt Weekly Quiz Competition

Probability Test - 43

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Probability Test - 43

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SHARING IS CARING

Question 1

$$\dfrac{5}{21}$$

$$\dfrac{3}{21}$$

$$\dfrac{4}{21}$$

$$\dfrac{1}{21}$$

Solution

Question 2

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

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

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

None

Solution

Question 3

$$\dfrac{7}{20}$$

$$\dfrac{8}{20}$$

$$\dfrac{6}{20}$$

$$\dfrac{9}{20}$$

Solution

Question 4

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

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

$$\dfrac{3}{13}$$

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

Solution

Question 5

Event $$A$$ and $$B$$ are mutually exclusive, exhaustive and complementary events

Event $$A$$ and $$B$$ are mutually exclusive and exhaustive events

Event $$A$$ and $$B$$ are mutually exclusive and complementary events

Event $$A$$ and $$B$$ are exhaustive and complementary events

Solution

Question 6

$$1$$

$$0$$

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

$$2$$

Solution

Question 7

$$\dfrac {7}{4}$$

$$-1$$

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

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

Solution

Question 8

0.61

0.16

0.14

0.6

Solution

Question 9

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

$$\dfrac{3}{10}$$

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

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

Solution

Question 10

$$1$$

$$0$$

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

$$-1$$

Solution

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