Bayes' Theorem

Q. In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is $\frac{1}{3}$ and the probability that he copies the answer is $\frac{1}{6}$. The probability that his answer is correct given that he copied it is $\frac{1}{8}.$ Find the probability that he knew the answer to the question, given that he answered it correctly.

1. $\frac{17}{21}$
2. $\frac{23}{27}$
3. $\frac{24}{29}$
4. $\frac{21}{23}$

Q.

If from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, one ball is drawn at random, then the probability that 2 white and 1 black ball will be drawn is

1. $\frac{13}{32}$

2. $\frac{1}{4}$

3. $\frac{1}{32}$

4. $\frac{3}{16}$

Q. There are three bags $B_1$, $B_2,$ and $B_3$.The bag $B_1$ contains 5 red and 5 green balls, $B_2$ contains 3 red and 5 green balls, $B_3$ contains 5 red and 3 green balls. Bags $B_1$, $B_2$ and $B_3$ have probabilities $\frac{3}{10},\frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following option is/are correct?

1. Probability that the chosen ball is green, given that the selected bag is $B_3,$ equals $\frac{3}{8}$
2. Probability that the chosen ball is green equals $\frac{39}{80}$
3. Probability that the selected bag is $B_3,$ given that the chosen ball is green, equals $\frac{5}{13}$
4. Probability that the selected bag is $B_3,$ and the chosen ball is green equals $\frac{3}{10}$

Q. $A$ can hit a target $4$ times out of $5$ shots, $B$ thrice in $4$ shots and $C$ twice in $3$ shots, independent of each other. They fire a volley. Two shots hit the target. Then, the probability that it is $C$ who has missed the target, is

1. $\frac{6}{13}$
2. $\frac{1}{5}$
3. $\frac{2}{5}$
4. $\frac{13}{30}$

Q. Consider four coins labelled as $1,2,3$ and $4.$ Suppose that the probability of obtaining a 'head' in a single toss of the $i^{th}$ coin is $\frac{i}{4},i=1,2,3,4.$ A coin is chosen uniformly at random and flipped. If it is given that the flip resulted in a 'head', and the conditional probability that the coin was labelled either $1$ or $2,$ is $p,$ then the value of $20p$ is

Q. A cricket club has $15$ members, of whom only $5$ can bowl. If the names of $15$ members are put into a box and $11$ are drawn at arandom, then the probability of getting an eleven containing at least $3$ bowlers is

1. $\frac{7}{13}$
2. $\frac{6}{13}$
3. $\frac{11}{15}$
4. $\frac{12}{13}$

Q. A letter is known to have come from either $\text{TATANAGAR}$ or $\text{CALCUTTA}$. On the envelope, just two consecutive letter $\text{TA}$ are visible. The probability that the letter has come from $\text{CALCUTTA}$ is

1. $\frac{4}{11}$
2. $\frac{1}{3}$
3. $\frac{5}{12}$
4. $\frac{1}{7}$

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. $\frac{75}{83}$

Q. Rohan throws an unbiased die. If he gets a $1,2,3,4,5$, he throws a coin and notes whether it is a head or a tail. But if he gets a $6$, he throws the coin four times and notes the number of heads. If he obtained exactly one head, then the probability that he threw a $6$ with the die is

(Assume the coin to be unbiased and each toss to be independent)

1. $\frac{4}{9}$
2. $\frac{1}{6}$
3. $\frac{1}{11}$
4. $\frac{5}{9}$

Q. There are three bags $B_1$, $B_2,$ and $B_3$.The bag $B_1$ contains 5 red and 5 green balls, $B_2$ contains 3 red and 5 green balls, $B_3$ contains 5 red and 3 green balls. Bags $B_1$, $B_2$ and $B_3$ have probabilities $\frac{3}{10},\frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following option is/are correct?

1. Probability that the chosen ball is green, given that the selected bag is $B_3,$ equals $\frac{3}{8}$
2. Probability that the chosen ball is green equals $\frac{39}{80}$
3. Probability that the selected bag is $B_3,$ given that the chosen ball is green, equals $\frac{5}{13}$
4. Probability that the selected bag is $B_3,$ and the chosen ball is green equals $\frac{3}{10}$

Q. Match the following by appropriately matching the lists based on the information given in Column I and Column II.
A bag contains some white and some black balls, all combinations being equally likely. The total number of balls in the bag is 12. Four balls are drawn at random from the bag at random without replacement.
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ a.\text{Probability that all the four balls are black is equal to}& p.\frac{14}{33}\\ b.\text{If the bag contains 10 black and 2 white balls, then}& \\ \text{the probability that all four balls are black is equal to}& q.\frac{1}{5}\\ c.\text{If all the four balls are black, then the probability}& \\ \text{that the bag contains 10 black balls is equal to}& r.\frac{70}{429}\end{array}$

1. $a\to r;b\to q;c\to p;$
2. $a\to r;b\to q;c\to p;$
3. $a\to q;b\to p;c\to r;$
4. $a\to r;b\to p;c\to q;$

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

Q. A software company employs 3 programmers A, B and C. For the duration of a particular project, A, B and C worked 30%, 50% and 20% respectively. The probability of python programming language being used by A, B and C is 5%, 10% and 15% respectively. Find the probability that a particular piece of code chosen randomly is written by A given that it is in python language.

1. 0.157
2. 0.186
3. 0.239
4. 0.175

Q. $A$ can hit a target $4$ times out of $5$ shots, $B$ thrice in $4$ shots and $C$ twice in $3$ shots, independent of each other. They fire a volley. Two shots hit the target. Then, the probability that it is $C$ who has missed the target, is

1. $\frac{1}{5}$
2. $\frac{13}{30}$
3. $\frac{6}{13}$
4. $\frac{2}{5}$

Q.

A bag contains some white and some black balls, all combinations of balls being equally likely. The total number of balls in the bag is 10. If three balls are drawn at random without replacement and all of them are found to be black, the probability that the bag contains 1 white and 9 black balls is

1. 14/55

2. 12/55

3. 2/11

4. 8/55

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 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. A box $B_1$ contains $1$ white ball, $3$ red balls, and $2$ black balls. Another box $B_2$ contains $2$ white balls, $3$ red balls, and $4$ black balls. A third box $B_3$ contains $3$ white balls, $4$ red balls, and $5$ black balls.
If $2$ balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these $2$ balls are drawn from box $B_2$ is

1. $\frac{116}{182}$
2. $\frac{126}{181}$
3. $\frac{65}{181}$
4. $\frac{55}{181}$

Q. There are 5 students who play guitar in class XI and 8 students who play guitar in class XII. Each class has 50 students. The probability of choosing class XI is $\frac{2}{5}$ and the probability of choosing class XII is $\frac{3}{5}$. A student is chosen and is found to be playing guitar. If the probability that the chosen student is from class XI is p, then the value of 34p is equal to

1. 10
2. 20
3. 5
4. 15

Q. Ram’s gardener is not dependable, the probability that he will forget to water the rose bush is$\frac{2}{3}$. The probability of its withering if watered is $\frac{1}{2}$and the probability of withering if not watered is $\frac{3}{4}$. Ram went out of station and upon returning, he finds that the rose bush has withered. If the probability that the gardener did not water the bush is p, then the value of 16p = ___

Q. An item is manufactured by three machines X, Y and Z. Out of the total number of items manufactured during a specified period, 40% are manufactured on X, 40% on Y and 20% on Z. 3% of the items produced on X and 2% of items produced on Y are defective, and 3% of these produced on Z is defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine Y?

1. $\frac{4}{13}$
2. $\frac{3}{7}$
3. $\frac{5}{7}$
4. $\frac{6}{13}$