Probability
235 Questions
Start JEE Mains Test
2021
Q151
JEE Mains
MCQ
14 Mar 2026
The coefficients a, b and c of the quadratic equation, ax2 + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :
A.
${1 \over {72}}$
B.
${5 \over {216}}$
C.
${1 \over {36}}$
D.
${1 \over {54}}$
2021
Q152
JEE Mains
MCQ
14 Mar 2026
The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :
A.
${{135} \over {{2^9}}}$
B.
${{65} \over {{2^8}}}$
C.
${{65} \over {{2^7}}}$
D.
${{35} \over {{2^7}}}$
2021
Q153
JEE Mains
MCQ
14 Mar 2026
An ordinary dice is rolled for a certain number of times. If the probability of getting an odd
number 2 times is equal to the probability of getting an even number 3 times, then the
probability of getting an odd number for odd number of times is :
A.
${5 \over {36}}$
B.
${3 \over {16}}$
C.
${1 \over 2}$
D.
${1 \over {32}}$
2021
Q154
JEE Mains
Numerical
14 Mar 2026
Let X be a random variable with distribution.
If the mean of X is 2.3 and variance of X is $\sigma$2, then 100 $\sigma$2 is equal to :
| x | $ - $2 | $ - $1 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| P(X = x) | ${1 \over 5}$ | a | ${1 \over 3}$ | ${1 \over 5}$ | b |
If the mean of X is 2.3 and variance of X is $\sigma$2, then 100 $\sigma$2 is equal to :
Correct Answer: 781
Explanation:
| x | $ - $2 | $ - $1 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| P(X = x) | ${1 \over 5}$ | a | ${1 \over 3}$ | ${1 \over 5}$ | b |
$\overline X $ = 2.3
$-$a + 6b = ${9 \over {10}}$ ..... (1)
$\sum {{P_i} = {1 \over 5} + a + {1 \over 3} + {1 \over 5} + b = 1} $
$a + b = {4 \over {15}}$ .... (2)
From equation (1) and (2)
$a = {1 \over {10}},b = {1 \over 6}$
${\sigma ^2} = \sum {{p_i}x_i^2 - {{(\overline X )}^2}} $
${1 \over 5}(4) + a(1) + {1 \over 3}(9) + {1 \over 5}(16) + b(36) - {(2.3)^2}$
$ = {4 \over 5} + a + 3 + {{16} \over 5} + 36b - {(2.3)^2}$
$ = 4 + a + 3 + 36b - {(2.3)^2}$
$ = 7 + a + 36b - {(2.3)^2}$
$ = 7 + {1 \over {10}} + 6 - {(2.3)^2}$
$ = 13 + {1 \over {10}} - {\left( {{{23} \over {10}}} \right)^2}$
$ = {{131} \over {10}} - {\left( {{{23} \over {10}}} \right)^2}$
$ = {{1310 - {{(23)}^2}} \over {100}}$
$ = {{1310 - 529} \over {100}}$
${\sigma ^2} = {{781} \over {100}}$
$100{\sigma ^2} = 781$
2021
Q155
JEE Mains
Numerical
14 Mar 2026
An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is p, then 98 p is equal to _____________.
Correct Answer: 28
Explanation:
I1 = first unit is functioning
I2 = second unit is functioning
P(I1) = 0.9, P(I2) = 0.8
P($\overline {{I_1}} $) = 0.1, P($\overline {{I_2}} $) = 0.2
$P = {{0.8 \times 0.1} \over {0.1 \times 0.2 + 0.9 \times 0.2 + 0.1 \times 0.8}} = {8 \over {28}}$
$98P = {8 \over {28}} \times 98 = 28$
I2 = second unit is functioning
P(I1) = 0.9, P(I2) = 0.8
P($\overline {{I_1}} $) = 0.1, P($\overline {{I_2}} $) = 0.2
$P = {{0.8 \times 0.1} \over {0.1 \times 0.2 + 0.9 \times 0.2 + 0.1 \times 0.8}} = {8 \over {28}}$
$98P = {8 \over {28}} \times 98 = 28$
2021
Q156
JEE Mains
Numerical
14 Mar 2026
The probability distribution of random variable X is given by :
Let p = P(1 < X < 4 | X < 3). If 5p = $\lambda$K, then $\lambda$ equal to ___________.
| X | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X) | K | 2K | 2K | 3K | K |
Let p = P(1 < X < 4 | X < 3). If 5p = $\lambda$K, then $\lambda$ equal to ___________.
Correct Answer: 30
Explanation:
$\sum {P(X) = 1 \Rightarrow k + 2k + 3} k + k = 1$
$ \Rightarrow k = {1 \over 9}$
Now, $p = P\left( {{{kx < 4} \over {X < 3}}} \right) = {{P(X = 2)} \over {P(X < 3)}} = {{{{2k} \over {9k}}} \over {{k \over {9k}} + {{2k} \over {9k}}}} = {2 \over 3}$
$ \Rightarrow p = {2 \over 3}$
Now, $5p = \lambda k$
$ \Rightarrow (5)\left( {{2 \over 3}} \right) = \lambda (1/9)$
$ \Rightarrow \lambda = 30$
$ \Rightarrow k = {1 \over 9}$
Now, $p = P\left( {{{kx < 4} \over {X < 3}}} \right) = {{P(X = 2)} \over {P(X < 3)}} = {{{{2k} \over {9k}}} \over {{k \over {9k}} + {{2k} \over {9k}}}} = {2 \over 3}$
$ \Rightarrow p = {2 \over 3}$
Now, $5p = \lambda k$
$ \Rightarrow (5)\left( {{2 \over 3}} \right) = \lambda (1/9)$
$ \Rightarrow \lambda = 30$
2021
Q157
JEE Mains
Numerical
14 Mar 2026
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 ______________.
Correct Answer: 4
Explanation:
P(Head) = ${1 \over 2}$
1 $-$ P(All tail) $\ge$ 0.9
$1 - {\left( {{1 \over 2}} \right)^n}$ $\ge$ 0.9
$ \Rightarrow {\left( {{1 \over 2}} \right)^n} \le {1 \over {10}}$
$\Rightarrow$ nmin = 4
1 $-$ P(All tail) $\ge$ 0.9
$1 - {\left( {{1 \over 2}} \right)^n}$ $\ge$ 0.9
$ \Rightarrow {\left( {{1 \over 2}} \right)^n} \le {1 \over {10}}$
$\Rightarrow$ nmin = 4
2021
Q158
JEE Mains
Numerical
14 Mar 2026
Let there be three independent events E1, E2 and E3. The probability that only E1 occurs is $\alpha$, only E2 occurs is $\beta$ and only E3 occurs is $\gamma$. Let 'p' denote the probability of none of events occurs that satisfies the equations
($\alpha$ $-$ 2$\beta$)p = $\alpha$$\beta$ and ($\beta$ $-$ 3$\gamma$)p = 2$\beta$$\gamma$. All the given probabilities are assumed to lie in the interval (0, 1).
Then, $\frac{Probability\ of\ occurrence\ of\ E_{1}}{Probability\ of\ occurrence\ of\ E_{3}} $ is equal to _____________.
($\alpha$ $-$ 2$\beta$)p = $\alpha$$\beta$ and ($\beta$ $-$ 3$\gamma$)p = 2$\beta$$\gamma$. All the given probabilities are assumed to lie in the interval (0, 1).
Then, $\frac{Probability\ of\ occurrence\ of\ E_{1}}{Probability\ of\ occurrence\ of\ E_{3}} $ is equal to _____________.
Correct Answer: 6
Explanation:
Let P(E1) = x, P(E2) = y and P(E3) = z
$\alpha $ = P$\left( {{E_1} \cap {{\overline E }_2} \cap {{\overline E }_3}} \right)$ = $P\left( {{E_1}} \right).P\left( {{{\overline E }_2}} \right).P\left( {{{\overline E }_3}} \right)$
$ \Rightarrow $ $\alpha $ = x(1 $-$ y) (1 $-$ z) ......(i)
Similarly
β = (1 – x).y(1 – z) ...(ii)
$\gamma $ = (1 – x)(1 – y).z ...(iii)
p = (1 – x)(1 – y)(1 – z) ...(iv)
From (i) and (iv)
${x \over {1 - x}} = {\alpha \over p}$
$ \Rightarrow $ x = ${\alpha \over {\alpha + p}}$
From (iii) and (iv)
${z \over {1 - z}} = {\gamma \over p}$
$ \Rightarrow $ z = ${\gamma \over {\gamma + p}}$
$ \therefore $ ${{P\left( {{E_1}} \right)} \over {P\left( {{E_3}} \right)}} = {x \over z} = {{{\alpha \over {\alpha + p}}} \over {{\gamma \over {\gamma + p}}}}$ $ = {{{{\gamma + p} \over \gamma }} \over {{{\alpha + p} \over \alpha }}} = {{1 + {p \over \gamma }} \over {1 + {p \over \alpha }}}$ ..(v)
Also given,
($\alpha$ $-$ 2$\beta$)p = $\alpha$$\beta$ $ \Rightarrow $ $\alpha $p = ($\alpha $ + 2p)$\beta $ ....(vi)
$\beta$ $-$ 3$\gamma$)p = 2$\beta$$\gamma$ $ \Rightarrow $ 3$\gamma $p = (p - 2$\gamma $)$\beta $ .....(vii)
From (vi) and (vii),
${\alpha \over {3\gamma }} = {{\alpha + 2p} \over {p - 2\gamma }}$
$ \Rightarrow $ p$\alpha $ - 6p$\gamma $ = 5$\gamma $$\alpha $
$ \Rightarrow $ ${p \over \gamma } - {{6p} \over \alpha } = 5$
$ \Rightarrow $ ${p \over \gamma } + 1 = 6\left( {{p \over \alpha } + 1} \right)$ ....(viii)
Now from (v) and (viii),
${{P\left( {{E_1}} \right)} \over {P\left( {{E_3}} \right)}}$ = 6
$\alpha $ = P$\left( {{E_1} \cap {{\overline E }_2} \cap {{\overline E }_3}} \right)$ = $P\left( {{E_1}} \right).P\left( {{{\overline E }_2}} \right).P\left( {{{\overline E }_3}} \right)$
$ \Rightarrow $ $\alpha $ = x(1 $-$ y) (1 $-$ z) ......(i)
Similarly
β = (1 – x).y(1 – z) ...(ii)
$\gamma $ = (1 – x)(1 – y).z ...(iii)
p = (1 – x)(1 – y)(1 – z) ...(iv)
From (i) and (iv)
${x \over {1 - x}} = {\alpha \over p}$
$ \Rightarrow $ x = ${\alpha \over {\alpha + p}}$
From (iii) and (iv)
${z \over {1 - z}} = {\gamma \over p}$
$ \Rightarrow $ z = ${\gamma \over {\gamma + p}}$
$ \therefore $ ${{P\left( {{E_1}} \right)} \over {P\left( {{E_3}} \right)}} = {x \over z} = {{{\alpha \over {\alpha + p}}} \over {{\gamma \over {\gamma + p}}}}$ $ = {{{{\gamma + p} \over \gamma }} \over {{{\alpha + p} \over \alpha }}} = {{1 + {p \over \gamma }} \over {1 + {p \over \alpha }}}$ ..(v)
Also given,
($\alpha$ $-$ 2$\beta$)p = $\alpha$$\beta$ $ \Rightarrow $ $\alpha $p = ($\alpha $ + 2p)$\beta $ ....(vi)
$\beta$ $-$ 3$\gamma$)p = 2$\beta$$\gamma$ $ \Rightarrow $ 3$\gamma $p = (p - 2$\gamma $)$\beta $ .....(vii)
From (vi) and (vii),
${\alpha \over {3\gamma }} = {{\alpha + 2p} \over {p - 2\gamma }}$
$ \Rightarrow $ p$\alpha $ - 6p$\gamma $ = 5$\gamma $$\alpha $
$ \Rightarrow $ ${p \over \gamma } - {{6p} \over \alpha } = 5$
$ \Rightarrow $ ${p \over \gamma } + 1 = 6\left( {{p \over \alpha } + 1} \right)$ ....(viii)
Now from (v) and (viii),
${{P\left( {{E_1}} \right)} \over {P\left( {{E_3}} \right)}}$ = 6
2021
Q159
JEE Mains
Numerical
14 Mar 2026
Let Bi (i = 1, 2, 3) be three independent events in a sample space. The probability that only B1 occur is $\alpha $, only B2 occurs is $\beta $ and only B3 occurs is $\gamma $. Let p be the probability that none of the events Bi occurs and these 4 probabilities satisfy the equations $\left( {\alpha - 2\beta } \right)p = \alpha \beta $ and $\left( {\beta - 3\gamma } \right)p = 2\beta \gamma $ (All the probabilities are assumed to lie in the interval (0, 1)).
Then ${{P\left( {{B_1}} \right)} \over {P\left( {{B_3}} \right)}}$ is equal to ________.
Then ${{P\left( {{B_1}} \right)} \over {P\left( {{B_3}} \right)}}$ is equal to ________.
Correct Answer: 6
Explanation:
Let x, y, z be probability of B1, B2, B3 respectively
$\alpha $ = P(B1 $ \cap $ $\overline {{B_2}} \cap \overline {{B_3}} $) = $P\left( {{B_1}} \right)P\left( {\overline {{B_2}} } \right)P\left( {\overline {{B_3}} } \right)$
$ \Rightarrow $ x(1 $-$ y)(1 $-$ z) = $\alpha$
Similarly, y(1 $-$ x)(1 $-$ z) = $\beta$
z(1 $-$ x)(1 $-$ y) = $\gamma$
and (1 $-$ x)(1 $-$ y)(1 $-$ z) = p
($\alpha$ $-$ 2$\beta$)p = $\alpha$$\beta$
(x(1 $-$ y)(1 $-$ z) $-$2y(1 $-$ x)(1 $-$ z)) (1 $-$ x)(1 $-$ y)(1 $-$ z) = xy(1 $-$ x)(1 $-$ y)(1 $-$ z)
x $-$ xy $-$ 2y + 2xy = xy
x = 2y ...... (1)
Similarly ($\beta$ $-$ 3$\gamma $)p = 2$\beta$$\gamma $
$ \Rightarrow $ y = 3z .... (2)
From (1) & (2)
x = 6z
Now
${x \over z} = 6$
$\alpha $ = P(B1 $ \cap $ $\overline {{B_2}} \cap \overline {{B_3}} $) = $P\left( {{B_1}} \right)P\left( {\overline {{B_2}} } \right)P\left( {\overline {{B_3}} } \right)$
$ \Rightarrow $ x(1 $-$ y)(1 $-$ z) = $\alpha$
Similarly, y(1 $-$ x)(1 $-$ z) = $\beta$
z(1 $-$ x)(1 $-$ y) = $\gamma$
and (1 $-$ x)(1 $-$ y)(1 $-$ z) = p
($\alpha$ $-$ 2$\beta$)p = $\alpha$$\beta$
(x(1 $-$ y)(1 $-$ z) $-$2y(1 $-$ x)(1 $-$ z)) (1 $-$ x)(1 $-$ y)(1 $-$ z) = xy(1 $-$ x)(1 $-$ y)(1 $-$ z)
x $-$ xy $-$ 2y + 2xy = xy
x = 2y ...... (1)
Similarly ($\beta$ $-$ 3$\gamma $)p = 2$\beta$$\gamma $
$ \Rightarrow $ y = 3z .... (2)
From (1) & (2)
x = 6z
Now
${x \over z} = 6$
2020
Q160
JEE Mains
MCQ
14 Mar 2026
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 :
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 :
A.
[0.35, 0.36]
B.
[0.20, 0.25]
C.
[0.25, 0.35]
D.
[0.36, 0.40]
2020
Q161
JEE Mains
MCQ
14 Mar 2026
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 :
A.
${{10} \over {99}}$
B.
${{5} \over {33}}$
C.
${{15} \over {101}}$
D.
${{5} \over {101}}$
2020
Q162
JEE Mains
MCQ
14 Mar 2026
In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws total a of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops as soon as either of the players wins. The probability of A winning the game is :
A.
${5 \over {6}}$
B.
${5 \over {31}}$
C.
${31 \over {61}}$
D.
${30 \over {61}}$
2020
Q163
JEE Mains
MCQ
14 Mar 2026
The probability that a randomly chosen 5-digit
number is made from exactly two digits is :
A.
${{150} \over {{{10}^4}}}$
B.
${{134} \over {{{10}^4}}}$
C.
${{121} \over {{{10}^4}}}$
D.
${{135} \over {{{10}^4}}}$
2020
Q164
JEE Mains
MCQ
14 Mar 2026
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 :
A.
${1 \over 8}$
B.
${1 \over 9}$
C.
${1 \over 4}$
D.
${1 \over 3}$
2020
Q165
JEE Mains
MCQ
14 Mar 2026
Let EC denote the complement of an event E.
Let E1
, E2
and E3
be any pairwise independent
events with P(E1) > 0
and P(E1 $ \cap $ E2 $ \cap $ E3) = 0.
Then P($E_2^C \cap E_3^C/{E_1}$) is equal to :
and P(E1 $ \cap $ E2 $ \cap $ E3) = 0.
Then P($E_2^C \cap E_3^C/{E_1}$) is equal to :
A.
$P\left( {E_3^C} \right)$ - P(E2)
B.
$P\left( {E_2^C} \right)$ + P(E3)
C.
$P\left( {E_3^C} \right)$ - $P\left( {E_2^C} \right)$
D.
P(E3) - $P\left( {E_2^C} \right)$
2020
Q166
JEE Mains
MCQ
14 Mar 2026
Box I contains 30 cards numbered 1 to 30 and
Box II contains 20 cards numbered 31 to 50. A
box is selected at random and a card is drawn
from it. The number on the card is found to be
a non-prime number. The probability that the
card was drawn from Box I is :
A.
${8 \over {17}}$
B.
${2 \over 3}$
C.
${2 \over 5}$
D.
${4 \over {17}}$
2020
Q167
JEE Mains
MCQ
14 Mar 2026
If 10 different balls are to be placed in 4 distinct
boxes at random, then the probability that two
of these boxes contain exactly 2 and 3 balls is :
A.
${{965} \over {{2^{11}}}}$
B.
${{965} \over {{2^{10}}}}$
C.
${{945} \over {{2^{11}}}}$
D.
${{945} \over {{2^{10}}}}$
2020
Q168
JEE Mains
MCQ
14 Mar 2026
A random variable X has the following
probability distribution :
Then P(X > 2) is equal to :
| X: | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X): | K2 | 2K | K | 2K | 5K2 |
Then P(X > 2) is equal to :
A.
${1 \over {6}}$
B.
${7 \over {12}}$
C.
${1 \over {36}}$
D.
${23 \over {36}}$
2020
Q169
JEE Mains
MCQ
14 Mar 2026
In a box, there are 20 cards, out of which 10
are lebelled as A and the remaining 10 are
labelled as B. Cards are drawn at random, one
after the other and with replacement, till a
second A-card is obtained. The probability that
the second A-card appears before the third
B-card is :
A.
${{13} \over {16}}$
B.
${{11} \over {16}}$
C.
${{15} \over {16}}$
D.
${{9} \over {16}}$
2020
Q170
JEE Mains
MCQ
14 Mar 2026
Let A and B be two events such that the
probability that exactly one of them occurs is ${2 \over 5}$ and the probability that A or B occurs is ${1 \over 2}$ ,
then the probability of both of them occur
together is :
A.
0.20
B.
0.02
C.
0.01
D.
0.10
2020
Q171
JEE Mains
MCQ
14 Mar 2026
Let A and B be two independent events such
that
P(A) = ${1 \over 3}$ and P(B) = ${1 \over 6}$.
Then, which of the following is TRUE?
P(A) = ${1 \over 3}$ and P(B) = ${1 \over 6}$.
Then, which of the following is TRUE?
A.
$P\left( {{A \over {A \cup B}}} \right) = {1 \over 4}$
B.
$P\left( {{A \over B}} \right) = {2 \over 3}$
C.
$P\left( {{{A'} \over {B'}}} \right) = {1 \over 3}$
D.
$P\left( {{A \over {B'}}} \right) = {1 \over 3}$
2020
Q172
JEE Mains
MCQ
14 Mar 2026
In a workshop, there are five machines and the probability of any one of them to be out of service on a day is ${{1 \over 4}}$
. If the probability that at most two machines will be out of service on the same day is ${\left( {{3 \over 4}} \right)^3}k$, then k is equal to :
A.
${{{17} \over 4}}$
B.
${{{17} \over 2}}$
C.
${{{17} \over 8}}$
D.
4
2020
Q173
JEE Mains
MCQ
14 Mar 2026
An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value of k when k
consecutive heads are obtained for k = 3, 4, 5, otherwise X takes the value -1. Then the expected
value of X, is :
A.
$ - {3 \over {16}}$
B.
$ - {1 \over 8}$
C.
${1 \over 8}$
D.
${3 \over {16}}$
2020
Q174
JEE Mains
Numerical
14 Mar 2026
In a bombing attack, there is 50% chance that
a bomb will hit the target. Atleast two
independent hits are required to destroy the
target completely. Then the minimum number
of bombs, that must be dropped to ensure that
there is at least 99% chance of completely
destroying the target, is __________.
Correct Answer: 11
Explanation:
Let n is total no. of bombs being dropped
at least 2 bombs should hit.
P(x > 2) $ \ge $ 0.99
$ \Rightarrow $ 1 - p(x < 2) $ \ge $ 0.99
$ \Rightarrow $ 1 - (p(x = 0) + p(x = 1)) $ \ge $ 0.99
$ \Rightarrow $ 1 - nC0${\left( {{1 \over 2}} \right)^0}{\left( {{1 \over 2}} \right)^n}$ - nC1.${\left( {{1 \over 2}} \right)^1}{\left( {{1 \over 2}} \right)^{n - 1}}$ $ \ge $ 0.99
$ \Rightarrow $ 1 - ${1 \over {{2^n}}}$ - ${n \over {{2^n}}}$ $ \ge $ ${{99} \over {100}}$
$ \Rightarrow $ ${1 \over {100}}$ $ \ge $ ${{n + 1} \over {{2^n}}}$
$ \Rightarrow $ 2n $ \ge $ 100(n + 1)
Now checking for value of n, we get
n = 11
at least 2 bombs should hit.
P(x > 2) $ \ge $ 0.99
$ \Rightarrow $ 1 - p(x < 2) $ \ge $ 0.99
$ \Rightarrow $ 1 - (p(x = 0) + p(x = 1)) $ \ge $ 0.99
$ \Rightarrow $ 1 - nC0${\left( {{1 \over 2}} \right)^0}{\left( {{1 \over 2}} \right)^n}$ - nC1.${\left( {{1 \over 2}} \right)^1}{\left( {{1 \over 2}} \right)^{n - 1}}$ $ \ge $ 0.99
$ \Rightarrow $ 1 - ${1 \over {{2^n}}}$ - ${n \over {{2^n}}}$ $ \ge $ ${{99} \over {100}}$
$ \Rightarrow $ ${1 \over {100}}$ $ \ge $ ${{n + 1} \over {{2^n}}}$
$ \Rightarrow $ 2n $ \ge $ 100(n + 1)
Now checking for value of n, we get
n = 11
2020
Q175
JEE Mains
Numerical
14 Mar 2026
The probability of a man hitting a target is ${1 \over {10}}$. The least number of shots required, so that the
probability of his hitting the target at least once is greater than ${1 \over {4}}$, is ____________.
Correct Answer: 3
Explanation:
We have, $1 - $(probability of all shots results in failure out of n trials) > ${1 \over 4}$
$ \Rightarrow 1 - {\left( {{9 \over {10}}} \right)^n} > {1 \over 4}$
$ \Rightarrow {3 \over 4} > {\left( {{9 \over {10}}} \right)^n} \Rightarrow n \ge 3$
$ \Rightarrow 1 - {\left( {{9 \over {10}}} \right)^n} > {1 \over 4}$
$ \Rightarrow {3 \over 4} > {\left( {{9 \over {10}}} \right)^n} \Rightarrow n \ge 3$
2019
Q176
JEE Mains
MCQ
14 Mar 2026
A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins
Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the
expected gain/loss (in Rs.) of the person is :
A.
${1 \over 4}$ loss
B.
${1 \over 2}$ gain
C.
${1 \over 2}$ loss
D.
2 gain
2019
Q177
JEE Mains
MCQ
14 Mar 2026
For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability
that the candidate solve any problem is ${4 \over 5}$
, then the probability that he is unable to solve less than two
problems is :
A.
${{164} \over {25}}{\left( {{1 \over 5}} \right)^{48}}$
B.
${{316} \over {25}}{\left( {{4 \over 5}} \right)^{48}}$
C.
${{201} \over 5}{\left( {{1 \over 5}} \right)^{49}}$
D.
${{54} \over 5}{\left( {{4 \over 5}} \right)^{49}}$
2019
Q178
JEE Mains
MCQ
14 Mar 2026
If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle
formed with these chosen vertices is equilateral is :
A.
${1 \over {10}}$
B.
${3 \over {10}}$
C.
${3 \over {20}}$
D.
${1 \over {5}}$
2019
Q179
JEE Mains
MCQ
14 Mar 2026
Let a random variable X have a binomial distribution with mean 8 and variance 4. If $P\left( {X \le 2} \right) = {k \over {{2^{16}}}}$, then k
is equal to :
A.
17
B.
1
C.
137
D.
121
2019
Q180
JEE Mains
MCQ
14 Mar 2026
Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is
more than 99% is :
A.
6
B.
5
C.
8
D.
7
2019
Q181
JEE Mains
MCQ
14 Mar 2026
Assume that each born child is equally likely to be a boy or a girl. If two families have two children each,
then the conditional probability that all children are girls given that at least two are girls is :
A.
${1 \over {10}}$
B.
${1 \over {17}}$
C.
${1 \over {11}}$
D.
${1 \over {12}}$
2019
Q182
JEE Mains
MCQ
14 Mar 2026
Four persons can hit a target correctly with
probabilities
${1 \over 2}$, ${1 \over 3}$, ${1 \over 4}$ and
${1 \over 8}$ respectively. if all hit
at the target independently, then the probability that
the target would be hit, is :
A.
${{25} \over {32}}$
B.
${{25} \over {192}}$
C.
${{1} \over {192}}$
D.
${{7} \over {32}}$
2019
Q183
JEE Mains
MCQ
14 Mar 2026
The minimum number of times one has to toss a
fair coin so that the probability of observing at least
one head is at least 90% is :
A.
2
B.
3
C.
4
D.
5
2019
Q184
JEE Mains
MCQ
14 Mar 2026
Let A and B be two non-null events such that
A $ \subset $ B . Then, which of the following statements
is always correct?
A.
P(A|B) = 1
B.
P(A|B) = P(B) – P(A)
C.
P(A|B) $ \le $ P(A)
D.
P(A|B) $ \ge $ P(A)
2019
Q185
JEE Mains
MCQ
14 Mar 2026
In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the students selected has opted neither for NCC
nor for NSS is :
A.
${1 \over 3}$
B.
${1 \over 6}$
C.
${2 \over 3}$
D.
${5 \over 6}$
2019
Q186
JEE Mains
MCQ
14 Mar 2026
In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :
A.
${{400} \over 3}$ loss
B.
0
C.
${{400} \over 9}$ loss
D.
${{400} \over 3}$ gain
2019
Q187
JEE Mains
MCQ
14 Mar 2026
In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :
A.
${{200} \over {{6^5}}}$
B.
${{225} \over {{6^5}}}$
C.
${{150} \over {{6^5}}}$
D.
${{175} \over {{6^5}}}$
2019
Q188
JEE Mains
MCQ
14 Mar 2026
Let S = {1, 2, . . . . . ., 20}. A subset B of S is said to be "nice", if the sum of the elements of B is 203. Then the probability that a randonly chosen subset of S is "nice" is :
A.
${5 \over {{2^{20}}}}$
B.
${7 \over {{2^{20}}}}$
C.
${4 \over {{2^{20}}}}$
D.
${6 \over {{2^{20}}}}$
2019
Q189
JEE Mains
MCQ
14 Mar 2026
A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then $\left( {{{mean\,\,of\,X} \over {s\tan dard\,\,deviation\,\,of\,X}}} \right)$ is equal to :
A.
4
B.
$3\sqrt 2 $
C.
${{4\sqrt 3 } \over 3}$
D.
$4\sqrt 3 $
2019
Q190
JEE Mains
MCQ
14 Mar 2026
Two integers are selected at random from the set {1, 2, ...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :
A.
${2 \over 5}$
B.
${1 \over 2}$
C.
${7 \over 10}$
D.
${3 \over 5}$
2019
Q191
JEE Mains
MCQ
14 Mar 2026
If the probability of hitting a target by a shooter, in any shot, is ${1 \over 3}$, then the minimum number of independent
shots at the target required by him so that the probability of hitting the target atleast once is greater than ${5 \over 6}$ is :
A.
4
B.
6
C.
5
D.
3
2019
Q192
JEE Mains
MCQ
14 Mar 2026
An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered 1, 2, 3, ……, 9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is :
A.
${{19} \over {36}}$
B.
${{15} \over {72}}$
C.
${{13} \over {36}}$
D.
${{19} \over {72}}$
2019
Q193
JEE Mains
MCQ
14 Mar 2026
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 :
A.
${{21} \over {49}}$
B.
${{27} \over {49}}$
C.
${{26} \over {49}}$
D.
${{32} \over {49}}$
2019
Q194
JEE Mains
MCQ
14 Mar 2026
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 :
A.
$25 \over 169$
B.
$49\over 169$
C.
$24 \over 169$
D.
$52 \over 169$
2018
Q195
JEE Mains
MCQ
14 Mar 2026
Let A, B and C be three events, which are pair-wise independent and $\overrightarrow E $ denotes the completement of an event E. If $P\left( {A \cap B \cap C} \right) = 0$ and $P\left( C \right) > 0,$ then $P\left[ {\left( {\overline A \cap \overline B } \right)\left| C \right.} \right]$ is equal to :
A.
$P\left( {\overline A } \right) - P\left( B \right)$
B.
$P\left( A \right) + P\left( {\overline B } \right)$
C.
$P\left( {\overline A } \right) - P\left( {\overline B } \right)$
D.
$P\left( {\overline A } \right) + P\left( {\overline B } \right)$
2018
Q196
JEE Mains
MCQ
14 Mar 2026
Two different families A and B are blessed with equal numbe of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family B is ${1 \over {12}},$ then the number of children in each family is :
A.
3
B.
4
C.
5
D.
6
2018
Q197
JEE Mains
MCQ
14 Mar 2026
A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and
this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at
random from the bag, then the probability that this drawn ball is red, is :
A.
${3 \over 4}$
B.
${3 \over 10}$
C.
${2 \over 5}$
D.
${1 \over 5}$
2018
Q198
JEE Mains
MCQ
14 Mar 2026
A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of 'p' is :
A.
${1 \over 5}$
B.
${1 \over 3}$
C.
${2 \over 5}$
D.
${1 \over 4}$
2018
Q199
JEE Mains
MCQ
14 Mar 2026
A box 'A' contains $2$ white, $3$ red and $2$ black balls. Another box 'B' contains $4$ white, $2$ red and $3$ black balls. If two balls are drawn at random, without eplacement, 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 'B' is :
A.
${9 \over {16}}$
B.
${7 \over {16}}$
C.
${9 \over {32}}$
D.
${7 \over {8}}$
2017
Q200
JEE Mains
MCQ
14 Mar 2026
Let E and F be two independent events. The probability that both E and F happen is ${1 \over {12}}$ and the probability that neither E nor F happens is ${1 \over {2}}$, then a value of ${{P\left( E \right)} \over {P\left( F \right)}}$ is :
A.
${4 \over 3}$
B.
${3 \over 2}$
C.
${1 \over 3}$
D.
${5 \over 12}$