Probability

Two dice are thrown and the sum of the numbers appearing on the dice is observed to be a multiple of 4 . If $p$ is the conditional probability that number 4 has appeared atleast once, then $3 p+2=$

The variance of a Poisson variate $X$ is 2 . Then, $P(X \geq 3)=$

A random variable $X$ has the following probability distribution
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline X=x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline P(X=x) & 0.15 & 0.23 & K & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \\ \hline \end{array} $$...

If the coefficients $a$ and $b$ of a quadratic expression $x^2+a x+b$ are chosen from the sets $A=\{3,4,5\}$ and $B=\{1,2,3,4\}$ respectively, then the probability that the equation $x^2+a x+b=0$ has real roots is

Let $B(\alpha, \beta, \gamma)$ represents that a bag $B$ contains $\alpha$ red balls, $\beta$ green balls and $\gamma$ blue balls. Given $B_1(2,3,2), B_2(3,2,2), B_3(2,2,3)$. A die is rolled. If the die shows up 2 or 3 or 5 , then a ball wi...

A person tossing a biased coin indefinitely wins the game by getting head for the first time. The probability that he wins the game in odd number of tosses is $3 / 4$. If 5 such coins are tossed at a time then the probability that head appe...

If the probability that an individual will suffer a reaction from an injection of a drug is 0.001 , then the probability that out of 2000 individuals having that injection, more than 2 individuals will suffer a reaction, is

If probability function of a discrete random variable $X$ is $P(X=r)=r / k, r=1,2,3,4,5$, then $P\left(X=2\right.$ or $\left.X=\frac{k}{3}\right)$, is

Consider the following statements
Assertion (A) If $P_1, P_2, P_3$ are probability of happening of three independent events, then probability of happening of atleast one of them is $1-\left[\left(1-P_1\right)\left(1-P_2\right)\left(1-P_3\ri...

A diagnostic test has the probability 0.95 of giving a positive result when applied to a person suffering from a certain disease and a probability 0.10 of giving a positive result when given to a non-sufferer. It is estimated that $0.5 \%$ ...

If a man throws a die until he gets a number bigger than 3 , then the probability that he gets a 5 in his last throw is

Let $X \sim B(n, p)$ with mean $\mu$ and variance $\sigma^2$. If $\mu=2 \sigma^2$ and $\mu+\sigma^2=3$, then $P(X \leq 3)=$

A target is to be destroyed in a bombing exercise and there is a $75 \%$ chance that a bomb will hit the target. Assuming that two direct hits are required to destroy the target completely, the minimum number of bombs to be dropped in order...

A number is selected at random from the set $\{1,2, \ldots \ldots ., 100\}$. Given that the number selected is divisible 2 , the probability that it is also divisible by 3 or 5 , is

Cards are drawn one after the other without replacement from a well shuffled pack of cards until and ace card appears. If the probability that exactly 5 cards are drawn before the first ace card appears is $\frac{4}{49}\left(\frac{p_1 \cdot...

If the roots of each of the equations $2 x^2+x-1=0$, $3 x^2-10 x+3=0$ and $6 x^2+11 x-2=0$ corresponds to probabilities of three events of a random experiment, then those events are

An observer counts 240 vehicles per hour at a specific location on a highway. Assuming that the arrival of vehicles at the location follows Poisson distribution, the probability that more than two vehicles arrive over a 30 sec time interval...

If the probability function of a random variable $X$ is given by $P(X=n)=\frac{k(n+1)}{3 n}$ for $n \in \mathbf{N} \cup\{0\}$ where $k$ is a constant, then $P(X<2)=$

An urn contains five balls. Two balls are drawn at random and they are found to be white. The probability that all the balls in the urn are white, is

If $E_1, E_2 \ldots, E_n$ are an independent events such that $P\left(E_r\right)=\frac{1}{1+r},(r=1,2, \ldots, n)$, then the probability that atleast one of $E_1, E_2, \ldots, E_n$ happens is

4-digit numbers are formed using the digits 4, 5, 6, 7, 8, 9 allowing repetition of the given digits. If a number is chosen at random from those numbers thus formed, then the probability that it is exactly divisible by 3 is

In a book consisting of 600 pages, there are 60 typographical errors. The probability that a randomly chosen page will contain at most two errors, is

If $20 \%$ of the bolts produced by a machine are defective then the probability that out of 4 bolts chosen at random, less than 2 bolts will be defective, is

Let $A$ and $B$ be not mutually exclusive events. If $P(A)=\frac{4}{9}, P(A \cap \bar{B})=\frac{3}{7}$ then $P\left(\frac{B}{A}\right)=$

Let $X$ and $Y$ be two events of a sample space such that $P(X)=\frac{1}{3}, P(X / Y)=\frac{1}{2}$ and $P(Y / X)=\frac{2}{5}$ then

If $A$ and $B$ are events of a sample space such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, then $P(\bar{A} \cap B)$ is

If $A_1, A_2, \ldots, A_{15}$ are the events of a random experiment, then which one of the following is true?

In an examination there are four Yes/No type of questions. The probability that the answer by the student to a question without guess to be correct is $2 / 3$. The probability that a student guesses a correct answer is $1 / 2$. A student wr...

Four boxes $A, B, C$ and $D$ contain 5000, 3000, 2000 and 1000 fuses respectively. The percentages of defective fuses in these boxes are $3 \%, 2 \%, 1 \%$ and $0.5 \%$ respectively. If a fuse selected at random from one of the boxes is fou...

In a hospital, on an average if there are 35 births in a weak, then the probability that there will be less than 3 births in a day, is

A die is thrown thrice. If getting 1 or 6 in a single throw is considered as success, then the variance of the number of successes is