Probability Test

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Probability Test - 57

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Question 1
1 / -0

A throws a coin $$3$$ times. If he get a head all three times, he is to get a reward of Rs.$$200$$, on the other hand if he does not get $$3$$ heads he is to loose Rs.40. He is expected to win Rs. ...............

A
Rs. $$50$$

B
Rs. $$(-10)$$

C
Rs. $$35$$

D
Rs. $$10$$

Solution

A throw of $$3$$ coins, total probability $$=8$$
$$\Rightarrow$$ Probability of getting head on all three throws $$=\dfrac{1}{8}$$
$$\Rightarrow$$ Probability of not getting head on all three throws $$=\dfrac{7}{8}$$
$$\therefore \dfrac{7}{8}\times40=35$$
$$\Rightarrow 35-(40+5)=Rs.(-10)$$
Hence, the answer is $$Rs.(-10).$$
Question 2
1 / -0

If $$\displaystyle P(A) = \frac{2}{3}$$, $$\displaystyle P(B) = \frac{4}{9}$$ and $$\displaystyle P(A \cap B) = \frac{14}{45}$$ then $$\displaystyle P(A' \cap B') $$ is greater than or equal to

A
$$\displaystyle \frac{34}{45}$$

B
$$\displaystyle \frac{1}{5}$$

C
$$\displaystyle -\frac{1}{5}$$

D
$$\displaystyle \frac{41}{45}$$

Solution
Question 3
1 / -0

The percentage of marks obtained by a student in the monthly unit tests are given.
Unit test $$1$$ $$2$$ $$3$$ $$4$$ $$5$$
Percentage of marks obtained $$60$$ $$75$$ $$55$$ $$72$$ $$85$$
Find the probability that the student gets Marks between $$80\%$$ and $$90\%$$.

A
$$\frac{4}{5}$$

B
$$\frac{1}{5}$$

C
$$\frac{3}{5}$$

D
$$\frac{2}{5}$$

Solution
Question 4
1 / -0

Let $${E}^{e}$$ denotes the complement of an event $$E$$. If $$E,F,G$$ are pairwise independent events with $$P(G)> 0$$ and $$P(E\cap F\cap G)=0$$. Then, $$P({ E }^{ e }\cap { F }^{ e }|G)$$ equals

A
$$P({ E }^{ e })+P({ F }^{ e })$$

B
$$P({ E }^{ e })-P({ F }^{ e })$$

C
$$P({ E }^{ e })-P(F)$$

D
$$P(E)-P({ F }^{ e })$$

Solution

Given: $$P(E^e)=1-P(E)$$ and E, F, G are pairwise independent events, P(G)>0, $$P(E\cap F\cap G)=0$$
To find: $$P(E^e\cap F^e|G)=?$$
Sol:
$$P(E^e\cap F^e|G)=\dfrac{P\left(E^e\cap F^e\cap G\right)}{P\left(G\right)}=\dfrac{P(E^e)P(F^e)P(G)}{P\left(G\right)}= P(E^e)P(F^e)\\\implies =P(E^e)[1-P(F)]=P(E^e)-P(E^e)P(F)\\\implies = P(E^e)-[1-P(E)]P(F)\\\implies = P(E^e)-P(F)+P(E)P(F)$$
E, F, G are independent and $$P(E\cap F\cap G)=0$$
$$\implies P(E)P(F)P(G)=0$$ but $$P(G)\ne 0$$
$$\implies P(E)P(F)=0$$
Hence $$P(E^e\cap F^e|G)= P(E^e)-P(F)$$
Question 5
1 / -0

The probability that $$A$$ can solve a problem is $$\dfrac {2 }{ 3}$$ and that $$B$$ can solve is $$\dfrac {3 }{4}$$. If both of them attempt the problem. What is probability that the problem act solved?

A
$$\dfrac {11 }{ 12}$$

B
$$\dfrac {7 }{ 12}$$

C
$$\dfrac {5 }{ 12}$$

D
$$\dfrac {9 }{ 12}$$

Solution

The problem won't be solved if both A and B.
dont solve which has probability of $$\left( 1-\cfrac { 2 }{ 3 } \right) \left( 1-\cfrac { 3 }{ 4 } \right) $$ $$=\quad \cfrac { 1 }{ 12 } $$
So, the probability that the problem is solve
= 1 - $$\ \cfrac { 1 }{ 12 } $$
$$=\ \cfrac { 11 }{ 12 } $$
Question 6
1 / -0

If the probability for A to fail in an examination is 0.8 and that for B is 0.7 , then the probability that atleast one of A or B fails, is

A
$$\frac { 4 }{ 25\\ \\ } $$

B
$$\frac { 6 }{ 25 } $$

C
$$\frac { 43 }{ 50 } $$

D
$$\frac { 47 }{ 50 } $$

Solution
Question 7
1 / -0

Let $$A,\ B,\ C$$ be pairwise independent even such that $$P(C)> 0$$ and $$P(A\cap B\cap C)=0$$, then $$P(A'\cap B'\cap C)$$ is equal to

A
$$P(A)-P(B')$$

B
$$P(A')+P(B')$$

C
$$P(A')-P(B')$$

D
$$P(A')-P(B)$$

Solution
Question 8
1 / -0

Two numbers $$b$$ and $$c$$ are chosen at random with replacement from the numbers $$1,\,2,\,3,\,4,\,5,\,6,\,7,\,8$$ and $$9$$.The probability that $${x}^{2}+bx+c>0$$ for all $$x\in R$$ is

A
$$\dfrac{17}{123}$$

B
$$\dfrac{32}{81}$$

C
$$\dfrac{82}{125}$$

D
$$\dfrac{45}{143}$$

Solution
Question 9
1 / -0

If $$A$$ and $$B$$ are two independent events such that $$P(A)=\dfrac{1}{3}$$ and $$P(B)=\dfrac{3}{4}$$, then $$p\left\{\dfrac{B}{(A\cup B)}\right\}$$

A
$$\dfrac{7}{10}$$

B
$$\dfrac{8}{10}$$

C
$$\dfrac{9}{10}$$

D
$$\dfrac{6}{10}$$

Solution
Question 10
1 / -0

If A and B are two events such that P (A)=$$\frac{1}{2}$$ and P (B) = $$\frac{2}{3}$$ , then which of the

A
$$P (A\cup B)\geq \frac{2}{3}$$

B
$$P (A\cap B^{-})\leq \frac{1}{3}$$

C
Both (A) & (B)

D
Only A

Solution