Probability Questions - JEE

NTA Test 15 (Numerical) A bag contains a mixed lot of red and blue balls. If two balls are drawn at random, the probability of drawing two blue balls and the probability of drawing one ball of each colour is six times the probability of drawing two blue balls. Then, the total number of red and blue balls in the bag is

NTA Test 16 (Numerical) If $a_1, a_2, b_1$ and $b_j$ take values in the set $\{1, -1, 0\}$, then the probability that the equation $a_1x + b_1y = 0$ satisfies $\frac{c}{p}$ (p & q are co-prime) then, $q - 2p$ is

Out of $3n$ consecutive integers, three are selected at random. Find the probability that their sum is divisible by 3.

For Problems 7–9: A JEE aspirant estimates that she will be successful with an 80% chance if she studies 10 hours per day, with a 60% chance if she studies 7 hours per day, and with a 40% chance if she studies 4 hours per day. She further believes that she will study 10 hours, 7 hours, and 4 hours per day with probabilities 0.1, 0.2, and 0.7, respectively.Given that she does not achieve success, the chance she studied for 4 hours is

$A$ speaks truth in 60% cases and $B$ speaks truth in 70% cases. The probability that they will say the same thing while describing a single event is

Two dice are thrown independently. Let A be the event that the number appeared on the 1st die is less than the number appeared on the 2nd die, B be the event that the number appeared on the 1st die is even and that on the second die is odd, and C be the event that the number appeared on the 1st die is odd and that on the 2nd die is even. Then

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce 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?

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 $T_1$) = 10$P$ (computer turns out to be defective given that it is produced in plant $T_2$), where $P(E)$ 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

Which of the following is not true?

Consider $f(x) = x^3 + ax^2 + bx + c$. Parameters a, b, c are chosen, respectively, by throwing a die three times. Then the probability that $f(x)$ is an increasing function is

A doctor is called to see a sick child. The doctor knows (prior to the visit) that 90% of the sick children in that neighbourhood are sick with the flue, denoted by F, while 10% are sick with the measles, denoted by M. The probability of having a rash for a child sick with the measles is 0.95. However, occasionally children with the flue also develop a rash with conditional probability 0.08. Upon examination the child, the doctor finds a rash, then the probability that the child has the measles, is

If the probability of hitting a target by a shooter, in any shot, is $1/3$, 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 $5/6$, is ______.

15 persons among whom are $A$ and $B$, sit down at random at a round table. The probability that there are 4 persons between $A$ and $B$ is ______ (up to four decimal places).

A cube painted red on all sides is cut into 125 equal small cubes. A small cube when picked is found to show red color on one of its faces. The probability that two more faces also show red color is :

Two numbers $x$ and $y$ are selected at random from $[0,1]$. Let $A$ be the event $x^2 + y^2 \leq \dfrac{1}{4}$. Then $P(A)$ is

A line is divided at random in three parts, what is the chance that they form the sides of a possible triangle?

816. Consider a family of $n$ children. Let two events $A$ and $B$ are defined as follows:$A$: is the event that the family has both boys and girls$B$: is the event that the family has atmost one girlIf the events $A$ and $B$ are independent, then find the value of $n$.[Note: Probability that a randomly selected child is a boy or girl is same.]

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

In ten trials of an experiment, if the probability of getting '4 successes' is maximum, then find the probability of failure in each trial.

If a and b are chosen randomly from the set consisting of numbers 1, 2, 3, 4, 5, 6 with replacement. Then the probability that $\lim_{x \to 0} \left[(a^x + b^x)/2\right]^{2/x} = 6$ is

If $x = 3^a + 3^{-a}$ is a positive integral value, then the probability that $x$ will have $3$ in its units place is

From the set $\{1, 2, 3, \ldots, 20\}$, two numbers are selected at random. The probability that their product is a perfect square is

A fair dice is rolled once. The probability that an odd number less than 5 turns up is ______, and the probability that a number greater than or equal to 3 turns up is ______.

A student appears for tests I, II, and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II, and III are, respectively, $p$, $q$, and 1/2. If the probability that the student is successful is 1/2, then $p(1 + q) =$

An urn contains 5 red and 5 black balls. A ball is drawn at random, its color is noted and is returned to the urn. Moreover, 2 additional balls of the color drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?

Two-thirds of the students in a class are boys and the rest are girls. It is known that probability of a girl getting a first class is 0.25 and that of a boy is 0.28. Then find the probability that a student chosen at random will get a first class.

Two friends Shubham and Ankur decided to discuss Probability over phone. At a random moment within period of 20 minutes, 'Shubham' telephones 'Ankur', waits for 2 minutes and then puts down the receiver. During the same 20 minutes 'Ankur' arrives home at a random moment, stays for 5 minutes and then leaves. Find the probability that the two will discuss the probability over phone.

A box contains 12 red and 6 white balls. Balls are drawn from the bag one at a time without replacement. If in 6 draws, there are at least 4 white balls, find the probability that exactly one white ball is drawn in the next two draws. (Binomial coefficients can be left as such.)

If $4$ distinct numbers are chosen randomly from the first $100$ natural numbers, then the probability that all $4$ of them are either divisible by $3$ or divisible by $5$ is

A fair coin is tossed $n$ times. Given that $P(\text{at least one head}) > \dfrac{9}{10}$, find the minimum value of $n$.

Two families each having 4 members are to be seated around a circular table with alternate red and blue chairs. If probability that members of same family are seated together is $p$, then $35p$ equals

The coefficients $a,b,c$ in the quadratic equation $ax^2+bx+c=0$ are chosen from the set $\{1,2,3,4,5,6,7,8\}$. The probability of this equation having repeated roots is:

A bag contains 3 white, 3 black and 2 red balls. One by one, three balls are drawn without replacing them. Find the probability that the third ball is red.

Two cards are drawn at random from a pack of 52 playing cards. If the odds against the event that one card is heart and other is an ace are $n:1$, then $n=$

Two dice are thrown simultaneously to get the co-ordinates on $x-y$ plane. Then the probability that this point lies inside or on the region bounded by $|x|+|y|=3$ is

If A and B are two events such that P$(A) = 0.7$,$P(B) = 0.4$and P(A$\cap$$B) = 0.5$, where B denotes the ¯ ¯¯¯ ¯ ¯¯¯ complement of B, then P (B | (A$\cup$B)) ¯ is equal:-

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 W . Let the probability P (W ) of choosing the n n word W satisfy P (W$) = 2P$(W n n$n-1$) ,$n > 1$.$\alpha$If$P(CDBEA) = 2$$\beta$,$\alpha$,$\beta$$\ in $N , then$\alpha$+$\beta$is equal to : _______$2 -1$

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

Question 1: If $P(A) = 0.8$, $P(B) = 0.5$, then $P(A \cap B)$ lies in the interval

A company has two plants $A$ and $B$ to manufacture motorcycles. 60% motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. 80% of the motorcycles manufactured at plant $A$ are rated of the standard quality, while 90% of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then $126p$ is

Example 29 (Assertion-Reason):Statement-1: A man P speaks truth with probability $p$ and another man Q speaks truth with probability $2p$. If P and Q contradict each other with probability $\frac{1}{2}$, then there are two values of $p$.Statement-2: A quadratic equation with real coefficients has two real roots.

One hundred identical coins, each with probability $p$ of showing up head, are tossed. If \(0

Three dice are rolled. If $P(\text{all different})=\dfrac{p}{q}$ (coprime), then $q-p$ is equal to

A fair $n$-faced die is rolled until a number $<n$ appears. If mean of tosses $=\dfrac{n}{9}$, then $n$ is equal to

$X\sim B(n,p)$, mean $-$ variance $=1$, $2P(X=2)=3P(X=1)$. Then $n^2P(X>1)$ is equal to

Neha lists all positive divisors of $(2010)^2$. She randomly selects 2 distinct divisors. Probability that exactly one is a perfect square is

From a group of 4 men and 3 women, a committee of 3 is chosen at random. The probability that the committee has exactly 2 men and 1 woman is

Let a random variable X take values 0, 1, 2, 3 with$P(X = 0) = P(X = 1) = p$,$P(X = 2) = P(X = 3)$and 2 E (X$) = 2E(X)$. Then the value of$8p - 1$is :

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is:

Probability Questions (959)