Probability

Let $X={(x,y)\inℤ\timesℤ:\frac{x^{2}}{8}+\frac{y^{2}}{20}<1andy^{2}<5x}$. Three distinct point $P,Q$ and $R$ are randomly chosen from $X$. Then the probability that $P,Q$ and $R$ form a triangle whose area is a positive…

Suppose that Box-I contains $8$ red, $3$ blue and $5$ green balls, Box-ll contains $24$ red, $9$ blue and $15$ green balls, Box-III contains $1$ blue, $12$ green and $3$ yellow balls, Box-IV contains $10$ green, $16$…

Let $n_{1}andn_{2}$ be the number of red and black balls, respectively, in box I. Let $n_{3}andn_{4}$ be the number of red and black balls, respectively, in box II. One of the two boxes, box I and box II, was selected…

Paragraph: A box $B_{1}$ contains $1$ white ball, $3$ red balls and $2$ black balls. Another box $B_{2}$ contains $2$ white balls, $3$ red balls and $4$ black balls. A third box $B_{3}$ contains $3$ white balls, $4$ red…

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $11 / 25$ and the probability of none of them occurring is $2 / 25$. If $P(T)$ denotes the probability of occurrence of the…

A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively, is received by station $A$ and then transmitted to station B. The probability of each station receiving the signal…

Consider the system of equations where $\quad a, b, c, d \in\{0,1\}$ Statement 1 The probability that the system of equations has a unique solution, is $3 / 8$. Statement 2 The probability that the system of equations…

Paragraph: There are $n$ urns each containing $(n+1)$ balls such that the ith urn contains $i$ white balls and $(n+1-i)$ red balls. Let $u_i$ be the event of selecting ith urn, $i=1,2,3, \ldots, n$ and $W$ denotes the…

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…

Three students $S_1, S_2$ and $S_1$ 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…

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…

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 $\frac{1}{3}$, then the probability that the experiment…