Bayes' Theorem

Q.

Rams gardener is not dependable. The probability that he will forget to water the rose bush is $\frac{2}{3}$. The rose bush is in questionable condition anyhow if watered the probability of its withering is $\frac{1}{2}$, if not watered, the probability of its withering is $\frac{3}{4}$. Ram went out of the station and upon returning, he finds that the rosebush has withered. The probability that the gardener did not water the bush is?

1. $\frac{1}{2}$

2. $\frac{1}{3}$

3. $\frac{3}{4}$

4. none of the above

Q.

In a group of $400$ people, $160$ are smokers and non-vegetarians; $100$ are smokers and vegetarians and the remaining $140$ are non-smokers and vegetarians. Their chances of getting a particular chest disorder are $35\%,20\%\mathrm{and}10\%$ respectively. A person is chosen from the group at random and is found to be suffering from chest disorder. The probability that the selected person is a smoker and non-vegetarian is:

1. $\begin{array}{l}\frac{7}{45}\end{array}$

2. $\begin{array}{l}\frac{8}{45}\end{array}$

3. $\begin{array}{l}\frac{14}{45}\end{array}$

4. $\begin{array}{l}\frac{28}{45}\end{array}$

Q.

In an entrance test that is graded on the basis of two examination, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7 the probability of passing at least one of them is 0.95. What is the probability of passing both?

Q.

A coin is tossed once. Find the probability of getting a head.

Q. Given the following statistics, what is the probability that a woman has cancer if she has a positive mammogram result?
1. 1% of women have cancer.
2. 90% of women who have cancer test positive on mammograms.
3. 8% of women will have false positives.

1. 20.5%
2. 8%
3. 10.2%
4. 12.2%

Q. A and B are two students. Their chances of solving a problem correctly are $\frac{1}{5}$ and $\frac{1}{6}$
respectively. If the probability of their making a common error is, $\frac{1}{20}$ and they obtain the same answer, then the probability of their answer to be correct is

1. $\frac{1}{2}$
2. $\frac{5}{11}$
3. $\frac{6}{11}$
4. 1

Q. Let $n_1$ and $n_2$ be the number of red and black balls, respectively in box I. Let $n_3$ and $n_4$ be the number of red and black balls, respectively in box II.
One of the two boxes, box I and box II was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II, is $\frac{1}{3},$ then the correct option(s) with the possible values of $n_1,n_2,n_3$ and $n_4$ is/are

1. $n_1=3,n_2=3,n_3=5,n_4=15$
2. $n_1=3,n_2=6,n_3=10,n_4=50$
3. $n_1=8,n_2=6,n_3=5,n_4=20$
4. $n_1=6,n_2=12,n_3=5,n_4=20$

Q. A person goes to office either by car, scooter, bus or train, the probability of which being $\frac{1}{7},\frac{3}{7},\frac{2}{7}and\frac{1}{7}$ respectively. The probabilities that he reaches the office late, if he takes a car, scooter, bus or train are $\frac{2}{9},\frac{1}{9},\frac{4}{9}and\frac{1}{9}$, respectively. Given that he reaches the office on time, then what is the probability that he travelled by car?

1. $\frac{1}{7}$
2. $\frac{2}{7}$
3. $\frac{2}{9}$
4. $\frac{4}{21}$

Q. In a group of $400$ people, $160$ are smokers and non-vegetarian; $100$ are smokers and vegetarian and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35\%,20\%$ and $10\%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:

1. $\frac{7}{45}$
2. $\frac{8}{45}$
3. $\frac{14}{45}$
4. $\frac{28}{45}$

Q. A letter is known to have come either from $TATANAGAR$ or from $CALCUTTA$. On the envelope, just two consecutive letter $TA$ are visible. What is the probability that the letter came from $TATANAGAR$.

1. $\frac{7}{11}$
2. $\frac{2}{3}$
3. $\frac{1}{5}$
4. $\frac{1}{2}$

Q. A card from a deck of $52$ cards is lost. From the remaining cards, two cards are drawn with replacement and found to be clubs, then the probability that

1. missing card is a club is $\frac{16}{523}$
2. missing card is a spade is $\frac{13}{51}$
3. missing card is a spade is $\frac{1}{4}$
4. missing card is a club is $\frac{1}{4}$

Q. A card from a pack of $52$ cards is lost. From the remaining cards, two cards are drawn and are found to be spades. The probability that the missing card is also a spade is $\frac{a}{b}$ then the value of $a+b$ is

Q.

A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces 20% and plant $T_2$ produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that P(computer turns out to be defective, given that it is produced in plant $T_1)$ = 10P (computer turns out to be defective, given that it is produced in plant $T_2$), where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then, the probability that it is produced in plant $T_2$, is ?

1. $\frac{36}{73}$

2. $\frac{47}{79}$

3. $\frac{78}{93}$

4. $\frac{75}{83}$