Axiomatic Approach

Q. If the probability of hitting a target by a shooter, in any shot, is $\frac{1}{3}$, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $\frac{5}{6}$, is :

1. $3$
2. $4$
3. $5$
4. $6$

Q.

The probability of A, B, and C solving a problem are $\frac{1}{2},\frac{2}{7},\frac{3}{8}$ respectively. If all the three try to solve the problem simultaneously, the probability that exactly one of them will solve it is

1. $\frac{25}{168}$

2. $\frac{25}{56}$

3. $\frac{20}{168}$

4. $\frac{30}{168}$

Q. Three critics review a book. Odds in favour of th ebook are $5:2,4:3$ and $3:4$, respectively, for the three critics. The probability that majority are in favor off the book is

1. $\frac{164}{343}$
2. $\frac{209}{343}$
3. $\frac{35}{49}$
4. $\frac{125}{343}$

Q. In a certain city only two newspapers A and B are published, it is known that 25% of the city population reads A and 20% reads B, while 8% reads both A and B. It is also known that 30% of those who read A but not B look into advertisements and 40% of those who read B but not A look into advertisements while 50% of those who read both A and B look into advertisements. If a person is chosen at random from the population, what is the percentage probability that he/she reads advertisements?

1. $\frac{139}{1000}$
2. $\frac{77}{250}$
3. $\frac{481}{2000}$
4. $\frac{61}{500}$

Q. A hunter's chance of shooting an animal at a distance $r$ is $\frac{a^2}{r^2}(r>a).$ He fires when $r=2a$ if he misses he reloads fires when $r=3a,4a,\cdots$. If he misses at a distance $na$, the animal escapes, then the odds against the hunter is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $\frac{n+1}{n-1}$
2. $\frac{n-1}{n+1}$
3. $\frac{n-1}{n}$
4. $\frac{n+1}{n}$

Q. Let A, B , and C be three independent events with $P\left(A\right)=\frac{1}{3},P\left(B\right)=\frac{1}{2},andP\left(C\right)=\frac{1}{4}.$ . The probability of exactly 2 of these events occurring, is equal to

1. $\frac{3}{4}$
2. $\frac{17}{24}$
3. $\frac{1}{4}$
4. $\frac{7}{24}$

Q. Let E and F be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$. If P(T) denotes the probability of occurrence of the event T, then

1. $P\left(E\right)=\frac{4}{5},P\left(F\right)=\frac{3}{5}$
2. $P\left(E\right)=\frac{1}{5},P\left(F\right)=\frac{2}{5}$
3. $P\left(E\right)=\frac{2}{5},P\left(F\right)=\frac{1}{5}$
4. $P\left(E\right)=\frac{3}{5},P\left(F\right)=\frac{4}{5}$

Q. The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least $0.96$, is

Q.

$A$ speaks truth in $75$% cases and $B$ in $80$% of the cases. In what percentage of the cases are they likely to contradict each other, narrating the same incident?

1. $5$%

2. $15\%$

3. $35\%$

4. $45\%$

Q. Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X=1) + P(X=2) equals:

1. $\frac{24}{169}$
2. $\frac{25}{169}$
3. $\frac{49}{169}$
4. $\frac{52}{169}$

Q.

The probabilities that A and B will die within a year are $p$ and $q$ respectively, then the probability that only one of them will be alive at the end of the year is

1. $p+q$

2. $p+q-2pq$

3. $p+q-pq$

4. $p+q+pq$

Q. An urn contains $5$ red and $2$ green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is :

1. $\frac{27}{49}$
2. $\frac{21}{49}$
3. $\frac{26}{49}$
4. $\frac{32}{49}$

Q.

A bag contains $5$ blue balls and unknown number of red balls, two balls are drawn at random. The probability of both of them are blue is $\frac{5}{14}$, then the number of red balls are

1. $3$

2. $2$

3. $4$

4. $5$

Q. Tweleve balls are distributed among three boxes. The probability that the first box contains three balls is

1. $\frac{110}{9}(\frac{2}{3}{)}^{10}$
2. $\frac{9}{110}(\frac{2}{3}{)}^{10}$
3. $\frac{{}^{12}{C}_3}{{12}^3}\times 2^9$
4. $\frac{{}^{12}{C}_3}{3^{12}}$

Q. ‘A’ speaks truth in 60% of the cases and ‘B’ in 90% of the cases. The percentage of cases they are likely to contradict each other in stating the same fact is

1. 0.42
2. 0.24
3. 0.5
4. 0.32

Q. There are three candidates A, B and C contesting in an election. The winning probability of A is 40% and winning probability of B is 30%. Find the probability that A or B will win the election.

1. 1
2. 0.4
3. 0.3
4. 0.7

Q. A signal which can be green or red with probability $\frac{4}{5}and\frac{1}{5}$ respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\frac{3}{4}$. If the signal received at station B is green, then the probability that the original signal green is

1. $\frac{3}{5}$
2. $\frac{6}{7}$
3. $\frac{20}{23}$
4. $\frac{9}{20}$

Q. There are two die $A$ and $B$ both having six faces. Die $A$ has three faces marked with $1$, two faces marked with $2$, and one face marked with $3$. Die $B$ has one face marked with $1$, two faces marked with $2$, and three faces marked with $3$. Both dices are thrown randomly once. If $E$ be the event of getting sum of the numbers appearing on top faces equal to $x$, let $P\left(E\right)$ be the probability of event $E$, then
$P\left(E\right)$ is minimum when $x$ equals to

1. $4$
2. $5$
3. $3$
4. $6$

Q. In a certain city only two newspapers A and B are published, it is known that 25% of the city population reads A and 20% reads B, while 8% reads both A and B. It is also known that 30% of those who read A but not B look into advertisements and 40% of those who read B but not A look into advertisements while 50% of those who read both A and B look into advertisements. If a person is chosen at random from the population, what is the percentage probability that he/she reads advertisements?

1. $\frac{139}{1000}$
2. $\frac{77}{250}$
3. $\frac{481}{2000}$
4. $\frac{61}{500}$

Q. From a bag containing 10 distinct balls, 6 balls are drawn simultaneously and replaced. Then 4 balls are drawn. The probability that exactly 3 balls are common to the drawings is

1. $\frac{9}{22}$
2. $\frac{6}{19}$
3. $\frac{8}{21}$
4. $\frac{5}{24}$

Q. A committee of 5 students is selected at random from a group consisting 10 boys and 5 girls. Given that there is at least one girl in the committee, calculate the probability that there are exactly 2 girls in the committee.

1. $\frac{400}{917}$
2. $\frac{143}{143}$
3. $\frac{400}{1001}$
4. $\frac{400}{991}$

Q. If $A$ and $B$ are two independent events such that $P\left(A\right)>0.5,P\left(B\right)>0.5$, $P(A\cap \overline{B})=\frac{3}{25},P(\overline{A}\cap B)=\frac{8}{25},$ then the value of $P(A\cap B)$ is

1. $\frac{12}{25}$
2. $\frac{18}{25}$
3. $\frac{14}{25}$
4. $\frac{24}{25}$

Q.

How do you solve probability problems $?$

Q. Three children are selected at random from a group of 6 boys and 4 girls. It is known that in this group exactly one girl and one boy belong to same parent. The probability that the selected group of children have no blood relations, is equal to

1. $\frac{1}{15}$
2. $\frac{13}{15}$
3. $\frac{14}{15}$
4. $\frac{2}{15}$

Q. The probability of success is p in one trial and the experiment be repeated n times. If the ratio of the probability of exactly r success to the probability of exactly r failures is independent of n and r, then p =

1. $\frac{1}{3}$
2. None of these
3. $\frac{1}{2}$
4. exists no such value of p

Q. The probabilities of two events A and B are 0.3 and 0.6 respectively. The probability that both A and B occur simultaneously is 0.18 Then, the Probability that neither A nor B occurs is

1. 0.1
2. 0.72
3. 0.28
4. 0.42

Q.

Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls and $U_2$ contains only 1 white ball. A fair coin is tossed. If the head appears then 1 ball is drawn at random from $U_1$and put into $U_2$. However, if the tail appears then 2 balls are drawn at random from $U_1$ and put into $U_2$. Now, 1 ball is drawn at random from $U_2$. Given that the drawn ball from $U_2$ is white, the probability that a head appeared on the coin is

1. $\frac{17}{23}$

2. $\frac{11}{23}$

3. $\frac{15}{23}$

4. $\frac{12}{23}$

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. Two persons A and B take turns in throwing a pair of dice. The first person to through 9 from both dice will be awarded the prize. If A throws first then the probability that B wins the game is

1. $\frac{9}{17}$
2. $\frac{8}{17}$
3. $\frac{8}{9}$
4. $\frac{1}{9}$

Q. A bag contains $4$ balls out of which some balls are white . If probability that a bag contains exactly $i$ ball is proportional to $i^2$. A ball is drawn at random from the bag and found to be white, then the probability that bag conatins exactly 2 white balls is $p$ then $25p$ is