Probability Questions (959)

In a certain town, 40% of the people have brown hair, 25% have brown eyes, and 15% have both brown hair and brown eyes. If a person selected at random from the town has brown hair, the probability that he also has brown eyes is

$P(X=x)=k(x+1)3^{-x}$, $x=0,1,2,\ldots$ If $k$ is a constant, then $P(X\geq2)$ is equal to

There are two vans each having numbered seats, 3 in the front and 4 at the back. There are 3 girls and 9 boys to be seated in the vans. The probability of 3 girls sitting together in a back row on adjacent seats, is

A die is rolled $n$ times. $P(\text{odd 7 times})=P(\text{odd 9 times})$. If $P(\text{even twice})=\dfrac{k}{2^{15}}$, then $k$ is equal to

Let A and B be two events such that \(P(A/B) = \frac{1}{6}\), \(P(A \cap B) = \frac{1}{4}\) and \(P(A) = \frac{1}{4}\), where \(\overline{A}\) stands for complement of event A. Then, events A and B are

$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of $5$ before $B$ throws a sum of $8$, and $B$ wins if he throws a sum of $8$ before $A$ throws a sum of $5$. The probability that $A$ wins if $A$ makes the first throw, is:

Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag $B_2$ is:

Three dice are rolled simultaneously. The probability that all three dice show the same number is

There are three bags $X$, $Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag $Z$ contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag $Y$, is:

6 married couples are together in a group of 12 people. If 4 people are selected at random, the probability that exactly one married couple is among the selected 4 is

A coin with $P(H)=\frac{3}{4}$ is tossed until H or 3 T appear. If $X$ = number of tosses, mean of $X$ is

Probability that \(A\) speaks truth is 4/5. A coin is tossed. \(A\) reports that a head appears. Find the probability that actually there was head.

Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac{3}{4}$ times their greatest possible product, is $\frac{m}{n}$, where $\gcd(m,n)=1$, then $n-m$ equals

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 $29/45$, then $n$ is equal to:

Let $A=[a_{ij}]$ be a square matrix of order $2$ with entries either $0$ or $1$. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:

Two small squares on a chess board are chosen at random. Then, the probability that they have a common side, is

A board has $16$ squares (in a $4\times 4$ grid). Out of these $16$ squares, two squares are chosen at random. The probability that they have no side in common is:

A candidate is given 50 problems. The probability of solving any problem is \(\dfrac{4}{5}\). The probability that he is unable to solve less than two problems is [JEE Main 2019]

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to:

If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A|B)$ and $P(B|A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\dfrac{P(\bar{A} \cup \bar{B})}{P(\bar{A} \cap \bar{B})}$ is:

A biased coin with $P(H)=\frac{1}{4}$ is tossed until head appears. If $N$ is the number of tosses and $P$(equation $64x^2+5Nx+1=0$ has no real roots) $=\dfrac{k}{?}$, then $k=27$.

Let $S=\{M=[a_{ij}],a_{ij}\in\{0,1,2\},1\leq i,j\leq2\}$ and $A=\{M\in S: M\text{ is invertible}\}$. Then $P(A)$ is equal to

Let $N$ be the sum of two dice. If $P(2^N<N!)=\dfrac{m}{n}$ (coprime), then $4m-3n$ is equal to

Let $a$, $b$ and $c$ denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that $ax^2+bx+c=0$ has all real roots is $\frac{m}{n}$, $\gcd(m,n)=1$, then $m+n$ is equal to

Let $S$ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $S$, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

In a certain city, only 2 newspapers A and B are published. It is known that 25% of the city population read A and 20% read B while 8% reads both A and B. It is also known that 30% of those who read A but not B look into advertisement 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. What is the percentage of the population who read an advertisement?

Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then, probability of event A is equal to

A random variable X has the following probability distribution: X: 1, 2, 3, 4, 5 and P(X): K, 2K, K²/2, 2K², 5K². Then P(X > 2) is equal to

Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag $A$, if the ball drawn is white, is:

The coefficients $a,b,c$ in the quadratic equation $ax^2+bx+c=0$ are from the set $\{1,2,3,4,5,6\}$. If the probability of this equation having one real root bigger than the other is $p$, then $216p$ equals:

Two numbers $k_{1}$ and $k_{2}$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_{1}}+i^{k_{2}},\,(i=\sqrt{-1})$ is non-zero, equals:

Three rotten apples are accidentally mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the variance of $x$ is

A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of $X$, then the value of $64(\mu + \sigma^2)$ is:

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the first exam is p. If he fails in one of the exams then the probability of his passing in the next exam is p/2, otherwise it remains the same. Find the probability that he will qualify.

Two integers $x$ and $y$ are chosen with replacement from the set $\{0,1,2,3,\ldots,10\}$. Then the probability that $|x-y|>5$ is:

One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is

Two cards are drawn one by one from a pack of cards. The probability of getting first card an ace and second a honoured one is (before drawing second card first card is not placed again in the pack)

A man throws a fair coin a number of times and gets 2 points for each head he throws and 1 point for each tail he throws. The probability that he gets exactly 6 points is

In a precision bombing attack, there is a 50% chance that any one bomb will strike the target. Two direct hits are required to destroy the target completely. The number of bombs which should be dropped to give a 99% chance or better of completely destroying the target can be

Two numbers \(x\) and \(y\) are selected at random from \([0, 1]\). The probability that \(|x - y| \leq \dfrac{1}{2}\) is

A cricket captain loses at least 7 out of 10 coin tosses. If the coin is fair, the probability of losing at least 7 tosses out of 10 is

A natural number is chosen at random from the first one hundred natural numbers. The probability that \(\frac{(x-20)(x-40)}{x-30}

If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is

A bag contains 3 red and 3 green balls and a person draws out 3 at random. He then drops 3 blue balls into the bag and again draws out 3 at random. The chance that the 3 later balls being all of different colors is

Three different dice are rolled three times. The Probability that they show different numbers only two times is:

A bag contains 20 coins. If the probability that the bag contains exactly 4 biased coin is 1/3 and that of exactly 5 biased coin is 2/3, then the probability that all the biased coin are sorted out from the bag in exactly 10 draws is

\(2^n\) players of equal strength are playing a knock out tournament. If they are paired at randomly in all rounds, find the probability that out of two particular players \(S_1\) and \(S_2\), exactly one will reach in semi-final \((n \in N,\ n \geq 2)\).

A candidate is given 50 problems. The probability of solving any problem is \(\dfrac{4}{5}\). The probability that he is unable to solve less than two problems is [JEE Main 2019]

A car is parked among N cars standing in a row, but not at either end. On his return, the owner finds that exactly 'r' of the N places are still occupied. The probability that the places neighboring his car are empty is

Let x and y be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25$. Then, the number of ways of choosing x and y, such that $x + y$ is divisible by 5, is ___.