Probability

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five is

If two fair dice are rolled, then the probability that the sum of the numbers on the upper faces is at least 9, is

 Suppose three coins are tossed simultaneously. If $X$ denotes the number of heads, then probability distribution of x is

If three fair coins are tossed, then variance of number of heads obtained, is

The p.m.f. of a random variable X is given by
$$\begin{aligned} \mathrm{P}[\mathrm{X}=x] & =\frac{\binom{5}{x}}{2^5}, \text { if } x=0,1,2,3,4,5 \\ & =0, \text { otherwise } \end{aligned}$$
Then which of the following is not correct?

The probability that a person who undergoes a bypass surgery will recover is 0.6 . the probability that of the six patients who undergo similar operations, half of them will recover is __________.

If $A$ and $B$ are two independent events such that $\mathrm{P}\left(\mathrm{A}^{\prime}\right)=0.75, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.65$ and $\mathrm{P}(\mathrm{B})=\mathrm{p}$, then value of $p$ is

One hundred identical coins, each with probability p , of showing up heads are tossed once. If $0<\mathrm{p}<1$ and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of $p$ is

A random variable x takes the values $0,1,2$, $3, \ldots$ with probability $\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{5}\right)^x$, where k is a constant, then $\mathrm{P}(\mathrm{X}=0)$ is

Let $\mathrm{A}, \mathrm{B}$ and C be three events, which are pairwise independent and $\bar{E}$ denote the complement of an event E . If $\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=0$ and $\mathrm{P}(\mathrm{C})>0$, then $\math...

A random variable $X$ has the following probability distribution

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidde...

 There are three events $\mathrm{A}, \mathrm{B}, \mathrm{C}$, one of which must and only one can happen. The odds are 8:3 against $\mathrm{A}, 5: 2$ against B and the odds against C is $43: 17 \mathrm{k}$, then value of k is

In a Binomial distribution consisting of 5 independent trials, probabilities of exactly 1 and 2 successes are 0.4096 and 0.2048 respectively, then the probability, of getting exactly 4 successes, is

In a game, 3 coins are tossed. A person is paid ₹ 100$, if he gets all heads or all tails; and he is supposed to pay ₹ 40 , if he gets one head or two heads. The amount he can expect to win/lose on an average per game in (₹) is

A random variable X has the following probability distribution

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;...

A random variable X assumes values $1,2,3, \ldots \ldots ., \mathrm{n}$ with equal probabilities. If $\operatorname{var}(X): E(X)=4: 1$, then $n$ is equal to

Minimum number of times a fair coin must be tossed, so that the probability of getting at least one head, is more than $99 \%$ is

$A$ and $B$ are independent events with $P(A)=\frac{3}{10}$, $\mathrm{P}(\mathrm{B})=\frac{2}{5}$, then $\mathrm{P}\left(\mathrm{A}^{\prime} \cup \mathrm{B}\right)$ has the value

Three fair coins numbered 1 and 0 are tossed simultaneously. Then variance Var (X) of the probability distribution of random variable \(\mathrm{X}\), where \(\mathrm{X}\) is the sum of numbers on the uppermost faces, is

The probability mass function of random variable X is given by
$$P[X=r]=\left\{\begin{array}{ll} \frac{{ }^n C_r}{32}, & n, r \in \mathbb{N} \\ 0, & \text { otherwise } \end{array} \text {, then } P[X \leq 2]=\right.$$

\(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are three events, one of which must and only one can happen. The odds in favor of \(\mathrm{A}\) are \(4: 6\), the odds against \(B\) are \(7: 3\). Thus, odds against \(\mathrm{C}\) are

In a Binomial distribution with \(\mathrm{n}=4\), if \(2 \mathrm{P}(\mathrm{X}=3)=3 \mathrm{P}(\mathrm{X}=2)\), then the variance is

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards, then mean of number of queens is

A problem in statistics is given to three students A, B and C. Their probabilities of solving the problem are \(\frac{1}{2}, \frac{1}{3}\) and \(\frac{1}{4}\) respectively. If all of them try independently, then the probability, that proble...

A man takes a step forward with probability 0.4 and backwards with probability 0.6 . The probability that at the end of eleven steps, he is one step away from the starting point is

Three critics review a book. For the three critics the odds in favour of the book are \(2: 5, 3: 4\) and \(4: 3\) respectively. The probability that the majority is in favour of the book, is given by

$$\text { If } f(x)= \begin{cases}3\left(1-2 x^2\right) & ; 0< x < 1 \\ 0 & ; \text { otherwise }\end{cases}$$ is a probability density function of \(\mathrm{X}\), then \(\mathrm{P}\left(\frac{1}{4} < x < \frac{1}{3}\right)\) is

For an initial screening of an entrance exam, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is \(\frac{4}{5}\), then the probability, that he is unable to solve less than two probl...

Two cards are drawn successively with replacement from well shuffled pack of 52 cards, then the probability distribution of number of queens is

A lot of 100 bulbs contains 10 defective bulbs. Five bulbs are selected at random from the lot and are sent to retail store. Then the probability that the store will receive at most one defective bulb is

There are 6 positive and 8 negative numbers. From these four numbers are chosen at random and multiplied. Then the probability, that the product is a negative number, is

The p.m.f. of a random variable \(\mathrm{X}\) is $$\mathrm{P}(x)=\left\{\begin{array}{cl}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)}, & x=1,2,3, \ldots \mathrm{n} \\ 0, & \text { otherwise }\end{array}\right.$$, then \(\mathrm{E}(\mathrm{X})\) is

A fair die with numbers 1 to 6 on their faces is thrown. Let \(\mathrm{X}\) denote the number of factors of the number, on the uppermost face, then the probability distribution of \(\mathrm{X}\) is

A card is drawn at random from a well shuffled pack of 52 cards. The probability that it is black card or face card is

If a continuous random variable \(\mathrm{X}\) has probability density function \(\mathrm{f}(x)\) given by
$$f(x)=\left\{\begin{array}{cl}
a x & , \text { if } 0 \leq x<1 \\
a & , \text { if } 1 \leq x<2 \\
3 a-a x & , \text { if } 2 \leq x...

Let \(\mathrm{X}\) be random variable having Binomial distribution \(B(7, p)\). If \(P[X=3]=5 P[X=4]\), then variance of \(\mathrm{X}\) is

A random variable \(X\) has the following probability distribution

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hid...

A random variable \(X\) has the probability distribution

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;paddin...

Two dice are rolled. If both dice have six faces numbered \(1,2,3,5,7,11\), then the probability that the sum of the numbers on upper most face is prime, is

\(\mathrm{A}\) and \(\mathrm{B}\) are independent events with \(\mathrm{P}(\mathrm{A})=\frac{1}{4}\) and \(\mathrm{P}(\mathrm{A} \cup \mathrm{B})=2 \mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A})\), then \(\mathrm{P}(\mathrm{B})\) is

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then the probability distribution of number of jacks is

An experiment succeeds twice as often as it fails. Then the probability, that in the next 6 trials there will be atleast 4 successes, is

The p.m.f of random variate \(\mathrm{X}\) is $$P(X)= \begin{cases}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)}, & x=1,2,3, \ldots \ldots, \mathrm{n} \\ 0, & \text { otherwise }\end{cases}$$ Then \(\mathrm{E}(\mathrm{X})=\)

Three fair coins with faces numbered 1 and 0 are tossed simultaneously. Then variance (X) of the probability distribution of random variable \(\mathrm{X}\), where \(\mathrm{X}\) is the sum of numbers on the upper most faces, is

A box contains 100 tickets numbered 1 to 100 . A ticket is drawn at random from the box. Then the probability, that number on the ticket is a perfect square, is

An irregular six faced die is thrown and the probability that, in 5 throws it will give 3 even numbers is twice the probability that it will give 2 even numbers. The number of times, in 6804 sets of 5 throws, you expect to give no even numb...

A binomial random variable \(\mathrm{X}\) satisfies \(9. p(X=4)=p(X=2)\) when \(n=6\). Then \(p\) is equal to

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with these three vertices is equilateral, equals ___________.

From a lot of 20 baskets, which includes 6 defective baskets, a sample of 2 baskets is drawn at random one by one without replacement. The expected value of number of defective basket is

Let a random variable \(\mathrm{X}\) have a Binomial distribution with mean 8 and variance 4. If \(\mathrm{P}(\mathrm{X} \leq 2)=\frac{\mathrm{K}}{2^{16}}\), then \(\mathrm{K}\) is