Bayes' Theorem

Q.

A boy read $\frac{3}{8}$ of book on one day and $\frac{4}{5}$ of the remainder on the other day.

If there were $30$ pages unread, how many pages did the book contain?

Q.

A laboratory blood test is 99% effective in detecting a certain diesease when it is fact present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e., if a healthy person is tested, then with probability 0.005, the test will imply he has the disease). If 0.1% of the population actually has the disease, what is the probability that a person has diease given that his test result is positive?

Q.

A biased coin has a property that probability of occurence of head is twice that of tail. What is the probability that if the coin is tossed twice then the head occurs atleast once?

1. 2×(2/3)²

2. 1/6

3. 1/3

4. (2/3)²

Q.

$X$ speaks truth in $60\%$ and $Y$ in $50\%$ of the cases. The probability that they contradict each other in narrating the same incident is

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

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

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

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

Q.

A shopkeeper sells three types of flower seeds $A_1,A_2$ and $A_3$ . They are sold as a mixture, where the proportions are 4 : 4 : 2, respectively. The germination rates of the three types of seeds are 45%, 60% and 35% . Calculate the probability

(i) of a randomly chosen seed to germinate.

(ii) that it will not germinate given that the seed is of type $A_3$.

(iii) that it is of the type $A_2$ given that a randomly chosen seed does not germinate.

Q. A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P(X\ge 5|X>2)$ is:

1. $\frac{5}{6}$
2. $\frac{125}{216}$
3. $\frac{11}{36}$
4. $\frac{25}{36}$

Q.

A certain type of missile hits the target with a probability of $p=0.3$. What is the least number of missiles that should be fired so that there is at least $80\%$ probability that the target is hit?

1. $5$

2. $6$

3. $7$

4. None of these

Q.

The mean and variance of a binomial distribution are 4 and 3 respectively, then the probability of getting exactly six successes in this distribution is:

1. 

2. 

3. 

4. 

Q. The content of $3$ urns $1,2,3$ are as follows : $1$ white, $2$ black, $3$ red balls; $2$ white, $1$ black, $1$ red balls; $4$ white, $5$ black, $3$ red balls. One urn is selected at random and two balls are drawn and these comes out to be white and red. Then the probability that they come from

1. $2nd\text{urn is}\frac{55}{118}$
2. $3rd\text{urn is}\frac{30}{118}$
3. $1st\text{urn is}\frac{33}{118}$
4. $3rd\text{urn is}\frac{59}{118}$

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. Three machines E₁, E₂, E₃ in a certain factory produce 50%, 25% and 25%, respectively, of the total daily output of electric bulbs. It is known that 4% of the tubes produced one each of the machines E₁ and E₂ are defective, and that 5% of those produced on E₃ are defective. If one tube is picked up at random from a day's production, then calculate the probability that it is defective.
[NCERT EXEMPLAR, CBSE 2015]

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 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. $\frac{12}{55}$
2. $\frac{2}{11}$
3. $\frac{8}{55}$
4. $\frac{14}{55}$

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 $P\left(E\right)$ 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.

A card from a pack of $52$cards is lost. from the remaining cards of pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

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 box ${}^{\prime }{A}^{\prime }$ contains $2$ white, $3$ red and $2$ black balls. Another box ${}^{\prime }{B}^{\prime }$ contains $4$ white, $2$ red and $3$ black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box ${}^{\prime }{B}^{\prime }$ is :

1. $\frac{9}{16}$
2. $\frac{7}{16}$
3. $\frac{7}{8}$
4. $\frac{9}{32}$

Q.

In answering a question on a multiple choice test a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4 What is the probability that a student knows the answer given that the answered it correctly?

Q. A bag contains 10 balls. 2 red, 3 blue and 5 black. Three are drawn at random. The probability that the three balls are of different colours is

1. $\frac{13}{120}$
2. $\frac{11}{120}$
3. $\frac{79}{120}$
4. $\frac{1}{4}$

Q.

A letter is known to have come either from $LONDON$ or $CLIFTON$; on the postmark only the two consecutive letters ON are eligible.

The probability that it came from $LONDON$ is

1. $\frac{5}{17}$

2. $\frac{12}{17}$

3. $\frac{17}{30}$

4. $\frac{3}{5}$

Q.

For two events $A$and $B$, if $P\left(A\right)=P(A/B)=\frac{1}{4}$ and $P(B/A)=\frac{1}{2}$, then

1. $A$and $B$ are independent

2. $P(A/B)=\frac{3}{4}$

3. $P(B/A)=\frac{1}{2}$

4. All of the above

Q.

Two groups are competing for the position on the board of directors of a corporation. The probability that the first and the second groups will win are 0.6 and 0.4, respectively. Further, if the first group wins the probability of introducing a new product is 0.7 and the corresponding probability is 0.3, if the second group wins. Find the probability that the new product introduce was by the second group.

Q.

Assume that the chances of a patient having a heart attack is$40\%$. It is also assumed that a meditation and yoga course reduces the risk of heart attack by $30\%$ and prescription of certain drug reduces its chances by $25\%$. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

Q.

A pack of playing cards was found to contain only $51$ cards. If the first $13$ cards which are examined are all red, then the probability that the missing is black is

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

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

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

4. $\frac{C_{13}^{25}}{C_{13}^{51}}$

Q. The probability that a randomly selected calculator from a store is of brand $r$ is proportional to $r(r=1,2,3\dots 6)$. Further, the probability of a calculator of brand $r$ being defective is $\frac{7-r}{21},$ $r=1,2,...6.$ If probability that a calculator randomly selected from the store being defective is $\frac{p}{q},$ where $p$ and $q$ are co-prime, then the value of $(p+q)$ is

Q.

A motorboat covers a given distance in $6$ hour moving downstream on a river. It covers the same distance in $10$ hour moving upstream. The time it takes to cover the same distance in still water is

1. $9$ hours

2. $7.5$ hours

3. $6.5$ hours

4. $8$ hours

Q. When 5-boys and 5-girls sit around a table the probability that no two girls come together

1. $\frac{1}{120}$
2. $\frac{1}{126}$
3. $\frac{3}{47}$
4. $\frac{4}{7}$

Q. A flashlight has $8$ batteries out of which $3$ are dead. If two batteries are selected without replacement and tested, the probability that both are dead is

Q.

In a knock-out chess tournament, eight players $P_1,P_2,.....P_8$ participated. It is known that whenever the players $P_i$ and $P_j$play, the players $P_i$will win j if $i<j.$ Assuming that the players are paired at random in each round, what is the probability that the player $P_4$ reaches the final?

1. $\frac{31}{35}$

2. $\frac{4}{35}$

3. $\frac{8}{35}$

4. none of the above.

Q. Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is hostler?