Probability

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let $E_1$ be the event that the ball chosen belonged to Box I and let $E_2$ be the event that the ball chosen belonged to Box II. Let $F_1$ be the event that the ball chosen is red and let $F_2$ be the event that the ball chosen is green.

Then which of the following statements is (are) TRUE ?

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let $X$ be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If $\alpha$ is the mean of the random variable $X$, then the value of $77\alpha$ is ________.

Let $a,b,c\in\{1,2,3,4\}$. If the probability, that $ax^2+2\sqrt2\,bx+c>0$ for all $x\in\mathbb R$, is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to ________.

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 $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to :

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 $\dfrac ab$, where $a,b\in\mathbb N$ and $\gcd(a,b)=1$, then $a+b$ is equal to ________.

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:

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 :

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 $\dfrac9{16}$, then N is equal to:

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 :

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 $\dfrac25$, $\dfrac15$ and $\dfrac25$. The probabilities that the candidate reaches late at the examination centre are $\dfrac15$, $\dfrac13$ and $\dfrac14$ 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 :

Three students $S_1, S_2$, and $S_3$ are given a problem to solve. Consider the following events:

• $U$: At least one of $S_1, S_2$, and $S_3$ can solve the problem,
• $V$: $S_1$ can solve the problem, given that neither $S_2$ nor $S_3$ can solve the problem,
• $W$: $S_2$ can solve the problem and $S_3$ cannot solve the problem,
• $T$: $S_3$ can solve the problem.

For any event $E$, let $P(E)$ denote the probability of $E$. If

$$P(U) = \frac12,\qquad P(V) = \frac{1}{10},\qquad \text{and}\qquad P(W) = \frac{1}{12},$$

then $P(T)$ is equal to

A factory has a total of three manufacturing units, $M_1$, $M_2$, and $M_3$, which produce bulbs independent of each other. The units $M_1$, $M_2$, and $M_3$ produce bulbs in the proportions of $2 : 2 : 1$, respectively. It is known that 20% of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by $M_1$, 15% are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by $M_2$ is $\dfrac25$.

If a bulb is chosen randomly from the bulbs produced by $M_3$, then the probability that it is defective is _____.

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is $\dfrac{1}{2}$. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is $\dfrac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question is

Let $X$ be a random variable, and let $P(X = x)$ denote the probability that $X$ takes the value $x$. Suppose that the points $(x, P(X = x))$, $x = 0, 1, 2, 3, 4$, lie on a fixed straight line in the $xy$-plane, and $P(X = x) = 0$ for all $x \in \mathbb{R} - \{0, 1, 2, 3, 4\}$. If the mean of $X$ is $\dfrac{5}{2}$, and the variance of $X$ is $\alpha$, then the value of $24\alpha$ is _____.

A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i = 1, 2, 3$, let $W_i$, $G_i$, and $B_i$ denote the events that the ball drawn in the $i^{\text{th}}$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P(W_1 \cap G_2 \cap B_3) = \dfrac{2}{5N}$ and the conditional probability $P(B_3 \mid W_1 \cap G_2) = \dfrac29$, then $N$ equals _____.

Let $X = \left\{(x, y) \in \mathbb{Z}\times\mathbb{Z} : \dfrac{x^2}{8} + \dfrac{y^2}{20} < 1 \text{ and } y^2 < 5x\right\}$. Three distinct points $P$, $Q$ and $R$ are randomly chosen from $X$. Then the probability that $P$, $Q$ and $R$ form a triangle whose area is a positive integer, is

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\dfrac13$, then the probability that the experiment stops with head is

Let $X$ be the set of all five digit numbers formed using 1, 2, 2, 2, 4, 4, 0. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of $38p$ is equal to _____.

Let $p_i$ be the probability that a randomly chosen point has $i$ many friends, $i = 0, 1, 2, 3, 4$. Let $X$ be a random variable such that for $i = 0, 1, 2, 3, 4$, the probability $P(X = i) = p_i$. Then the value of $7E(X)$ is _____.