Probability · JEE Main previous year questions

If a point $A(x,y)$ lies in the region bounded by the y-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is :

1. A.$\frac{1}{6}$
2. B.$\frac{5}{6}$
3. C.$\frac{2}{3}$
4. D.$\frac{6}{7}$

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is :

1. A.$\frac{275}{6^5}$
2. B.$\frac{36}{5^4}$
3. C.$\frac{181}{5^5}$
4. D.$\frac{46}{6^4}$

Let ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ be two events such that the conditional probabilities $P({\mathrm{E}}_1|{\mathrm{E}}_2)=\frac{1}{2}$, $P({\mathrm{E}}_2|{\mathrm{E}}_1)=\frac{3}{4}$ and $P({\mathrm{E}}_1\cap {\mathrm{E}}_2)=\frac{1}{8}$. Then :

1. A.$P({\mathrm{E}}_1\cap {\mathrm{E}}_2)=P({\mathrm{E}}_1)\cdot P({\mathrm{E}}_2)$
2. B.$P({\mathrm{E}}_1^{\prime }\cap {\mathrm{E}}_2^{\prime })=P({\mathrm{E}}_1^{\prime })\cdot P({\mathrm{E}}_2)$
3. C.$P({\mathrm{E}}_1\cap {\mathrm{E}}_2^{\prime })=P({\mathrm{E}}_1)\cdot P({\mathrm{E}}_2)$
4. D.$P({\mathrm{E}}_1^{\prime }\cap {\mathrm{E}}_2)=P({\mathrm{E}}_1)\cdot P({\mathrm{E}}_2)$

A biased die is marked with numbers $2,4,8,16,32,32$ on its faces and the probability of getting a face with mark $n$ is $\frac{1}{n}$. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is $48$, is :

1. A.$\frac{7}{2^{11}}$
2. B.$\frac{7}{2^{12}}$
3. C.$\frac{3}{2^{10}}$
4. D.$\frac{13}{2^{12}}$

A random variable X has the following probability distribution : $$\begin{array}{llllll}\text{X}& 0& 1& 2& 3& 4\\ \text{P(X)}& \mathrm{k}& 2\mathrm{k}& 4\mathrm{k}& 6\mathrm{k}& 8\mathrm{k}\end{array}$$ The value of $\mathrm{P}(1<\mathrm{X}<4\mid \mathrm{X}\le 2)$ is equal to :

1. A.$\frac{4}{7}$
2. B.$\frac{2}{3}$
3. C.$\frac{3}{7}$
4. D.$\frac{4}{5}$

In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability $\frac{3}{4}$ and the remaining 6 questions correctly with probability $\frac{1}{4}$. If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is $\frac{27k}{4^{10}}$, then k is equal to ____.

1. A.$\frac{2}{7}$
2. B.$\frac{1}{18}$
3. C.$\frac{1}{7}$
4. D.$\frac{1}{9}$

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

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

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations $x+y+z=5$ $x+2y+3z=\mu$ $x+3y+\lambda z=1$ is constructed. If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then :

1. A.$p=\frac{1}{6}$ and $q=\frac{1}{36}$
2. B.$p=\frac{5}{6}$ and $q=\frac{5}{36}$
3. C.$p=\frac{5}{6}$ and $q=\frac{1}{36}$
4. D.$p=\frac{1}{6}$ and $q=\frac{5}{36}$

A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. the smallest value of n, so that the probability of guessing at least 'n' correct answers is less than $\frac{1}{2}$, is :

1. A.5
2. B.6
3. C.3
4. D.4

Let 9 distinct balls be distributed among 4 boxes, $B_1,B_2,B_3$ and $B_4$. If the probability that $B_3$ contains exactly 3 balls is $k{\left(\frac{3}{4}\right)}^9$ then k lies in the set :

1. A.$\{x\in R:|x-3|<1\}$
2. B.$\{x\in R:|x-2|\le 1\}$
3. C.$\{x\in R:|x-1|<1\}$
4. D.$\{x\in R:|x-5|\le 1\}$

Let X be a random variable such that the probability function of a distribution is given by $P(X=0)=\frac{1}{2},P(X=j)=\frac{1}{3^j}(j=1,2,3,\dots ,\infty )$. Then the mean of the distribution and P(X is positive and even) respectively are :

1. A.$\frac{3}{8}$ and $\frac{1}{8}$
2. B.$\frac{3}{4}$ and $\frac{1}{8}$
3. C.$\frac{3}{4}$ and $\frac{1}{9}$
4. D.$\frac{3}{4}$ and $\frac{1}{16}$

A fair coin is tossed n-times such that the probability of getting at least one head is at least 0.9. Then the minimum value of n is ______________.

Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is :

1. A.$\frac{1}{66}$
2. B.$\frac{1}{11}$
3. C.$\frac{1}{9}$
4. D.$\frac{2}{11}$

The probability of selecting integers $a\in [-5,30]$ such that $x^2+2(a+4)x-5a+64>0$, for all $x\in \mathrm{R}$, is :

1. A.$\frac{7}{36}$
2. B.$\frac{2}{9}$
3. C.$\frac{1}{6}$
4. D.$\frac{1}{4}$

Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :

1. A.$\frac{40}{243}$
2. B.$\frac{128}{625}$
3. C.$\frac{80}{243}$
4. D.$\frac{32}{625}$

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

1. A.$\frac{39}{50}$
2. B.$\frac{3}{4}$
3. C.$\frac{22}{425}$
4. D.$\frac{52}{867}$

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is :

1. A.$\frac{15}{2^8}$
2. B.$\frac{15}{2^{12}}$
3. C.$\frac{15}{2^{13}}$
4. D.$\frac{15}{2^{14}}$

A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is :

1. A.$\frac{1}{7}$
2. B.$\frac{4}{7}$
3. C.$\frac{6}{7}$
4. D.$\frac{3}{7}$

Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :

1. A.$\frac{2}{9}$
2. B.$\frac{1}{5}$
3. C.$\frac{122}{297}$
4. D.$\frac{97}{297}$

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. A.$\frac{14}{45}$
2. B.$\frac{8}{45}$
3. C.$\frac{7}{45}$
4. D.$\frac{28}{45}$

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is :

1. A.$\frac{10}{99}$
2. B.$\frac{5}{33}$
3. C.$\frac{15}{101}$
4. D.$\frac{5}{101}$

The probabilities of three events A, B and C are given by $P(A)=0.6$, $P(B)=0.4$ and $P(C)=0.5$. If $P(A\cup B)=0.8$, $P(A\cap C)=0.3$, $P(A\cap B\cap C)=0.2$, $P(B\cap C)=\beta$ and $P(A\cup B\cup C)=\alpha$, where $0.85\le \alpha \le 0.95$, then $\beta$ lies in the interval :

1. A.$[0.35,0.36]$
2. B.$[0.20,0.25]$
3. C.$[0.25,0.35]$
4. D.$[0.36,0.40]$

A dice is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is :

1. A.$\frac{1}{8}$
2. B.$\frac{1}{9}$
3. C.$\frac{1}{4}$
4. D.$\frac{1}{3}$

The probability that a randomly chosen 5-digit number is made from exactly two digits is :

1. A.$\frac{150}{10^4}$
2. B.$\frac{134}{10^4}$
3. C.$\frac{121}{10^4}$
4. D.$\frac{135}{10^4}$