Probability · JEE Main previous year questions

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are $\frac{2}{5}$, $\frac{1}{5}$ and $\frac{2}{5}$. The probabilities that the candidate reaches late at the examination centre are $\frac{1}{5}$, $\frac{1}{3}$ and $\frac{1}{4}$ if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is :

1. A.$\frac{11}{37}$
2. B.$\frac{12}{37}$
3. C.$\frac{13}{37}$
4. D.$\frac{14}{37}$

A bag contains $(N+1)$ coins $-$ N fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is $\frac{9}{16}$, then N is equal to:

1. A.5
2. B.7
3. C.8
4. D.9

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

1. A.$\frac{63}{925}$
2. B.$\frac{17}{231}$
3. C.$\frac{16}{231}$
4. D.$\frac{64}{925}$

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:

1. A.$\frac{7}{10}$
2. B.$\frac{10}{17}$
3. C.$\frac{12}{19}$
4. D.$\frac{7}{19}$

The probabilities that players A and B of a team are selected for the captaincy for a tournament are $0.6$ and $0.4$, respectively. If A is selected the captain, the probability that the team wins the tournament is $0.8$ and if B is selected the captain, the probability that the team wins the tournament is $0.7$. Then the probability, that the team wins the tournament, is :

1. A.$0.74$
2. B.$0.76$
3. C.$0.72$
4. D.$0.78$

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$, then $96p$ is equal to _________.

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\frac{a}{b}$, where $a,b\in \mathbb{N}$ and $\gcd (a,b)=1$, then $a+b$ is equal to ____.

One mole of an alkane (x) requires 8 mole oxygen for complete combustion. Sum of number of carbon and hydrogen atoms in the alkane (x) is ________.

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is $\frac{m}{n}$, $\gcd (m,n)=1$, then $m+n$ is equal to :

1. A.53
2. B.55
3. C.107
4. D.105

A bag contains $10$ balls out of which $k$ are red and $(10-k)$ are black, where $0\le k\le 10$. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains $1$ red and $9$ black balls is:

1. A.$\frac{7}{110}$
2. B.$\frac{7}{11}$
3. C.$\frac{7}{55}$
4. D.$\frac{14}{55}$

The probability distribution of a random variable X is given below : $$\begin{array}{ccccccccc}\text{X}& 4k& \frac{30}{7}k& \frac{32}{7}k& \frac{34}{7}k& \frac{36}{7}k& \frac{38}{7}k& \frac{40}{7}k& 6k\\ \text{P(X)}& \frac{2}{15}& \frac{1}{15}& \frac{2}{15}& \frac{1}{5}& \frac{1}{15}& \frac{2}{15}& \frac{1}{5}& \frac{1}{15}\end{array}$$ If $\text{E(X)}=\frac{263}{15}$, then $\text{P(X}<20)$ is equal to :

1. A.$\frac{3}{5}$
2. B.$\frac{14}{15}$
3. C.$\frac{8}{15}$
4. D.$\frac{11}{15}$

From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is

1. A.$\frac{73}{10^8}$
2. B.$\frac{67}{10^8}$
3. C.$\frac{7}{10^7}$
4. D.$\frac{81}{10^8}$

From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the probability that $a-b⩾10$ is $\frac{m}{n}$, $\gcd (m,n)=1$, then $m+n$ is equal to ____ .

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A . Then a ball is randomly drawn from the bag A . If the probability, that the ball drawn is white, is $\frac{p}{q}$, $\gcd (p,q)=1$, then $p+q$ is equal to

1. A.24
2. B.22
3. C.23
4. D.21

Two distinct numbers $a$ and $b$ are selected at random from $1,2,3,\dots ,50$. The probability, that their product $ab$ is divisible by 3 , is

1. A.$\frac{272}{1225}$
2. B.$\frac{561}{1225}$
3. C.$\frac{664}{1225}$
4. D.$\frac{8}{25}$

If a random variable $x$ has the probability distribution $$\begin{array}{ccccccccc}x& 0& 1& 2& 3& 4& 5& 6& 7\\ \mathrm{P}(x)& 0& 2\mathrm{k}& \mathrm{k}& 3\mathrm{k}& 2{\mathrm{k}}^2& 2\mathrm{k}& {\mathrm{k}}^2+\mathrm{k}& 7{\mathrm{k}}^2\end{array}$$ then $\mathrm{P}(3<x\le 6)$ is equal to

1. A.0.34
2. B.0.64
3. C.0.22
4. D.0.33

If $A$ and $B$ are two events such that $P(A)=0.7$, $P(B)=0.4$ and $P(A\cap \overline{B})=0.5$, where $\overline{B}$ denotes the complement of $B$, then $P\left(B\mid (A\cup \overline{B})\right)$ is equal to

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

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is

1. A.$\frac{129}{182}$
2. B.$\frac{17}{26}$
3. C.$\frac{19}{26}$
4. D.$\frac{103}{182}$

A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $\frac{11}{50}$, then n is equal to ____ .

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by ${\mathrm{W}}_n$. Let the probability $\mathrm{P}\left({\mathrm{W}}_n\right)$ of choosing the word ${\mathrm{W}}_n$ satisfy $\mathrm{P}\left({\mathrm{W}}_n\right)=2\mathrm{P}\left({\mathrm{W}}_{n-1}\right),n>1$. If $\mathrm{P}(\mathrm{CDBEA})=\frac{2^{\alpha }}{2^{\beta }-1}$, $\alpha ,\beta \in \mathbb{N}$, then $\alpha +\beta$ is equal to : ____

If the probability that the random variable $X$ takes the value $x$ is given by $P(X=x)=k(x+1)3^{-x},x=0,1,2,3\dots$, where $k$ is a constant, then $P(X\ge 3)$ is equal to

1. A.$\frac{1}{9}$
2. B.$\frac{8}{27}$
3. C.$\frac{7}{27}$
4. D.$\frac{4}{9}$

Three distinct numbers are selected randomly from the set $\{1,2,3,\dots ,40\}$. If the probability, that the selected numbers are in an increasing G.P., is $\frac{m}{n}$, $\gcd (m,n)=1$, then $m+n$ is equal to _________ .

Given three identical bags each containing 10 balls, whose colours are as follows : $$\begin{array}{lccc}& \text{Red}& \text{Blue}& \text{Green}\\ \text{Bag I}& 3& 2& 5\\ \text{Bag II}& 4& 3& 3\\ \text{Bag III}& 5& 1& 4\end{array}$$ A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is p and if the ball is Green, the probability that it is from bag III is $q$, then the value of $\left(\frac{1}{p}+\frac{1}{q}\right)$ is:

1. A.6
2. B.9
3. C.7
4. D.8

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is $\frac{29}{45}$, then $n$ is equal to:

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

Two number ${\mathrm{k}}_1$ and ${\mathrm{k}}_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of ${\mathrm{i}}^{{\mathrm{k}}_1}+{\mathrm{i}}^{{\mathrm{k}}_2}$, ($\mathrm{i}=\sqrt{-1}$) is non-zero, equals

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