Probability and statistics

1. In a given day in the rainy season, it may rain 70% of the time. If it rains, chances that a village fair make a loss on that day is 80%. However, if it does not rain, chances that fair will make a loss on that day is only 10%. If the fair has not made a loss on a given day in the rainy season, what is the probability that it has not rained on that day? A. $$\frac{3}{{10}}$$ B. $$\frac{9}{{11}}$$ C. $$\frac{{14}}{{17}}$$ D. $$\frac{{27}}{{41}}$$

$$rac{3}{{10}}$$
$$rac{9}{{11}}$$
$$rac{{14}}{{17}}$$
$$rac{{27}}{{41}}$$

Detailed SolutionIn a given day in the rainy season, it may rain 70% of the time. If it rains, chances that a village fair make a loss on that day is 80%. However, if it does not rain, chances that fair will make a loss on that day is only 10%. If the fair has not made a loss on a given day in the rainy season, what is the probability that it has not rained on that day? A. $$\frac{3}{{10}}$$ B. $$\frac{9}{{11}}$$ C. $$\frac{{14}}{{17}}$$ D. $$\frac{{27}}{{41}}$$

2. An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3? A. 0.453 B. 0.468 C. 0.485 D. 0.492

0.453
0.468
0.485
0.492

Detailed SolutionAn unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3? A. 0.453 B. 0.468 C. 0.485 D. 0.492

3. If X is a continuous random variable whose probability density function is given by \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {{\text{K}}\left( {5{\text{x}} – 2{{\text{x}}^2}} \right)}&{0 \leqslant {\text{x}} \leqslant 2} \\ 0&{{\text{otherwise}}} \end{array}} \right.\] then P(x > 1) is A. $$\frac{3}{{14}}$$ B. $$\frac{4}{5}$$ C. $$\frac{{14}}{{17}}$$ D. $$\frac{{17}}{{28}}$$

$$rac{3}{{14}}$$
$$rac{4}{5}$$
$$rac{{14}}{{17}}$$
$$rac{{17}}{{28}}$$

Detailed SolutionIf X is a continuous random variable whose probability density function is given by \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {{\text{K}}\left( {5{\text{x}} – 2{{\text{x}}^2}} \right)}&{0 \leqslant {\text{x}} \leqslant 2} \\ 0&{{\text{otherwise}}} \end{array}} \right.\] then P(x > 1) is A. $$\frac{3}{{14}}$$ B. $$\frac{4}{5}$$ C. $$\frac{{14}}{{17}}$$ D. $$\frac{{17}}{{28}}$$

4. A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes: i. Head, ii. Head, iii. Head, iv. Head. The probability of obtaining a ‘Tail’ when the coin is tossed again is A. 0 B. $$\frac{1}{2}$$ C. $$\frac{4}{5}$$ D. $$\frac{1}{5}$$

0
$$rac{1}{2}$$
$$rac{4}{5}$$
$$rac{1}{5}$$

Detailed SolutionA fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes: i. Head, ii. Head, iii. Head, iv. Head. The probability of obtaining a ‘Tail’ when the coin is tossed again is A. 0 B. $$\frac{1}{2}$$ C. $$\frac{4}{5}$$ D. $$\frac{1}{5}$$

5. In a service centre, cars arrive according to Poisson distribution with a mean of two cars per hour. The time of servicing a car is exponential with a mean of 15 minutes. The expected waiting time (in minute) in the queue is A. 10 B. 15 C. 25 D. 30

10
15
25
30

Detailed SolutionIn a service centre, cars arrive according to Poisson distribution with a mean of two cars per hour. The time of servicing a car is exponential with a mean of 15 minutes. The expected waiting time (in minute) in the queue is A. 10 B. 15 C. 25 D. 30

6. A box contains 4 red balls and 6 black balls. Three balls are selected randomly from the box one after another, without replacement. The probability that the selected set contains one red ball and two black balls is A. $$\frac{1}{{20}}$$ B. $$\frac{1}{{12}}$$ C. $$\frac{3}{{10}}$$ D. $$\frac{1}{2}$$

$$rac{1}{{20}}$$
$$rac{1}{{12}}$$
$$rac{3}{{10}}$$
$$rac{1}{2}$$

Detailed SolutionA box contains 4 red balls and 6 black balls. Three balls are selected randomly from the box one after another, without replacement. The probability that the selected set contains one red ball and two black balls is A. $$\frac{1}{{20}}$$ B. $$\frac{1}{{12}}$$ C. $$\frac{3}{{10}}$$ D. $$\frac{1}{2}$$

7. Two n bit binary strings, S1 and S2 are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is A. $$\frac{{{}^{\text{n}}{{\text{C}}_{\text{d}}}}}{{{2^{\text{n}}}}}$$ B. $$\frac{{{}^{\text{n}}{{\text{C}}_{\text{d}}}}}{{{2^{\text{d}}}}}$$ C. $$\frac{{\text{d}}}{{{2^{\text{n}}}}}$$ D. $$\frac{1}{{{2^{\text{d}}}}}$$

$$rac{{{}^{ ext{n}}{{ ext{C}}_{ ext{d}}}}}{{{2^{ ext{n}}}}}$$
$$rac{{{}^{ ext{n}}{{ ext{C}}_{ ext{d}}}}}{{{2^{ ext{d}}}}}$$
$$rac{{ ext{d}}}{{{2^{ ext{n}}}}}$$
$$rac{1}{{{2^{ ext{d}}}}}$$

Detailed SolutionTwo n bit binary strings, S1 and S2 are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is A. $$\frac{{{}^{\text{n}}{{\text{C}}_{\text{d}}}}}{{{2^{\text{n}}}}}$$ B. $$\frac{{{}^{\text{n}}{{\text{C}}_{\text{d}}}}}{{{2^{\text{d}}}}}$$ C. $$\frac{{\text{d}}}{{{2^{\text{n}}}}}$$ D. $$\frac{1}{{{2^{\text{d}}}}}$$

8. An unbiased coin is tossed five times. The outcome of each toss is either a head or a tail. The probability of getting at least one head is A. $$\frac{1}{{32}}$$ B. $$\frac{{13}}{{32}}$$ C. $$\frac{{16}}{{32}}$$ D. $$\frac{{31}}{{32}}$$

$$rac{1}{{32}}$$
$$rac{{13}}{{32}}$$
$$rac{{16}}{{32}}$$
$$rac{{31}}{{32}}$$

Detailed SolutionAn unbiased coin is tossed five times. The outcome of each toss is either a head or a tail. The probability of getting at least one head is A. $$\frac{1}{{32}}$$ B. $$\frac{{13}}{{32}}$$ C. $$\frac{{16}}{{32}}$$ D. $$\frac{{31}}{{32}}$$

9. A box contains 10 screws, 3 of which are defective. Two screws are drawn at random with replacement. The probability that none of the two screws is defective will be A. 100% B. 50% C. 49% D. None of these

100%
50%
49%
None of these

Detailed SolutionA box contains 10 screws, 3 of which are defective. Two screws are drawn at random with replacement. The probability that none of the two screws is defective will be A. 100% B. 50% C. 49% D. None of these

10. From a pack of regular playing cards, two cards are drawn at random. What is the probability that both cards will be Kings, if first card in NOT replaced? A. $$\frac{1}{{26}}$$ B. $$\frac{1}{{52}}$$ C. $$\frac{1}{{169}}$$ D. $$\frac{1}{{221}}$$

$$rac{1}{{26}}$$
$$rac{1}{{52}}$$
$$rac{1}{{169}}$$
$$rac{1}{{221}}$$

Detailed SolutionFrom a pack of regular playing cards, two cards are drawn at random. What is the probability that both cards will be Kings, if first card in NOT replaced? A. $$\frac{1}{{26}}$$ B. $$\frac{1}{{52}}$$ C. $$\frac{1}{{169}}$$ D. $$\frac{1}{{221}}$$

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