CBSE Questions for Class 12 Commerce Applied Mathematics Standard Probability Distributions Quiz 9 - MCQExams.com

If, in a Poisson distribution $$P(X= 0)=k$$ then the variance is: 
  • $$e^{\lambda}$$
  • $$\log \dfrac{1}{k}$$
  • $$\dfrac{1}{k}$$
  • $$\log k$$
Let a random variable X have a binomial distribution with mean $$8$$ and variance $$4$$. If $$P(x\leq 2)=\dfrac{k}{2^{16}}$$, then k is equal to?
  • $$17$$
  • $$1$$
  • $$121$$
  • $$137$$
A die is thrown $$7$$ times. The chance that an odd number turns up exactly 4 times, is given by
  • $$\dfrac{1}{{128}}$$
  • $$\dfrac{{35}}{{128}}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{{31}}{{128}}$$
If n different apples are to be distributed among m children, find the chance that the particular child receives p apples.
  • $$\displaystyle\frac{^{ n }{ C_p }(m-1)^{n-p }}{m^n}$$
  • $$\displaystyle\frac{^{ n }{ C_p }(m-1)^{n-p }}{n^m}$$
  • $$\displaystyle\frac{^{ n }{ P_p }(m-1)^{n-p }}{n^m}$$
  • $$\displaystyle\frac{^{ n }{ P_p }(m-1)^{n-p }}{m^n}$$
The minimum number of times one has to toss fair coins so that probability of observing at least one head is at least $$90$$% is:
  • $$5$$
  • $$3$$
  • $$2$$
  • $$4$$
The mean and variance of a random variable $$X$$ having a binomial distribution are $$6$$ and $$3$$ respectively. The probability of variable $$X$$ less than $$2$$ is
  • $$\cfrac { 13 }{ 2048 } $$
  • $$\cfrac { 13 }{ 4096 } $$
  • $$\cfrac { 15 }{ 4096 } $$
  • $$\cfrac { 25 }{ 2048 } $$

A coin is biased so that the head is $$3$$ times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.
  • $$P(T=0)=\dfrac{11}{16}$$
  • $$P(T=1)=\dfrac{6}{16}$$
  • $$P(T=2)=\dfrac{13}{16}$$
  • none of these
In a workshop, there are five machines and the probability of any one of them to be out of service on day is $$\dfrac{1}{4}$$. If the probability that at most two machines will be out of service on the same day is $$\left(\dfrac{3}{4} \right)^3 k$$, then k is equal to :
  • $$4$$
  • $$\dfrac{17}{2}$$
  • $$\dfrac{17}{8}$$
  • $$\dfrac{17}{4}$$
The mean and variance of a binomial distribution are $$8$$ and $$4$$ respectively. What is $$(X = 1)$$ ?
  • $$\dfrac{1}{2^8}$$
  • $$\dfrac{1}{2^{12}}$$
  • $$\dfrac{1}{2^6}$$
  • $$\dfrac{1}{2^4}$$
  • $$\dfrac{1}{2^5}$$
A poison variate X satisfies $$P(X - 1) = P (X = 2)$$
$$P(X = 6)$$ is equal to 
  • $$\dfrac{4}{45} e^{-2}$$
  • $$\dfrac{1}{45} e^{-1}$$
  • $$\dfrac{1}{9} e^{-2}$$
  • $$\dfrac{1}{4} e^{-2}$$
  • $$\dfrac{1}{45} e^{-2}$$
An unbiased coin is tossed $$5$$ times. Suppose that a variable $$X$$ is assigned the value $$k$$ when $$k$$ consecutive heads are obtained for $$k=3,4,5,$$ otherwise $$X$$ takes the value $$-1$$. The the expected value of $$X$$, is :
  • $$\dfrac{1}{8}$$
  • $$\dfrac{3}{16}$$
  • $$-\dfrac{1}{8}$$
  • $$-\dfrac{3}{16}$$
If the mean and variance of a binomial variate $$X$$ are $$\dfrac {7}{3}$$ and $$\dfrac {14}{9}$$ respectively. Then probability that $$X$$ takes value $$6$$ or $$7$$ is equal to 
  • $$\dfrac {1}{729}$$
  • $$\dfrac {5}{729}$$
  • $$\dfrac {7}{729}$$
  • $$\dfrac {13}{729}$$
Suppose $$X$$ is a binomial variate $$B(5,p)$$ and $$P(X=2)=P(X=3)$$, then $$p$$ is equal to 
  • $$1/2$$
  • $$1/3$$
  • $$1/4$$
  • $$1/5$$
A coin is tossed $$5$$ times. What is the probability that tail appears an odd number of times?
  • $$\cfrac{3}{5}$$
  • $$\cfrac{2}{15}$$
  • $$\cfrac{1}{2}$$
  • $$\cfrac{1}{3}$$
A die is thrown $$5$$ times. If getting an odd number is a success, then what is the probability of getting atleast $$4$$ successes?
  • $$\cfrac{4}{5}$$
  • $$\cfrac{7}{16}$$
  • $$\cfrac{3}{16}$$
  • $$\cfrac{3}{20}$$
A coin is tossed $$5$$ times. What is the probability that head appears an even number of times?
  • $$\cfrac{2}{5}$$
  • $$\cfrac{3}{5}$$
  • $$\cfrac{4}{15}$$
  • $$\cfrac{1}{2}$$
If $$X$$ follows binomial distribution with parameters $$n=8$$ and $$p =1/2$$ then $$p\left(|x-4| < 2\right)=$$
  • $$121/128$$
  • $$119/128$$
  • $$117/128$$
  • $$115/128$$
A fair coin is tossed $$6$$ times. What is the probability of getting at least $$3$$ heads?
  • $$\cfrac{11}{16}$$
  • $$\cfrac{21}{32}$$
  • $$\cfrac{1}{18}$$
  • $$\cfrac{3}{64}$$
The probability of the safe arrival of one ship out of $$5$$ is $$\cfrac{1}{5}$$. What is the probability of the safe arrival of at least $$3$$ ships?
  • $$\cfrac{1}{31}$$
  • $$\cfrac{3}{52}$$
  • $$\cfrac{181}{3125}$$
  • $$\cfrac{184}{3125}$$
$$8$$ coins are tossed simultaneously. The probability of getting at least $$6$$ heads is 
  • $$\cfrac{7}{64}$$
  • $$\cfrac{57}{64}$$
  • $$\cfrac{37}{256}$$
  • $$\cfrac{249}{256}$$
Suppose  random variable X follows the binomial distribution with parameters n and p, where $$0<p<1$$. If $$P(x=r)/P(x=n-r)$$ is independent of n and r, then p equals
  • 1/2
  • 1/3
  • 1/5
  • 1/7
2K coins (K is an integer) each with probability P(O<P<1) of getting head are tossed together. If the probability of getting K heads is equal to the probability of getting K+1 heads, the value of P is
  • $$\dfrac{K}{2K+1}$$
  • $$\dfrac{K+1}{2K}$$
  • $$\dfrac{K+1}{2K+1}$$
  • $$\dfrac{2K}{K+1}$$
A discrete random variable $$X$$ can take all possible integer values from $$1$$ to $$K$$, each with a probability $$\dfrac{1}{k}$$. Its variance is
  • $$\dfrac{k^{2}}{4}$$
  • $$\dfrac{(k+1)^{2}}{4}$$
  • $$\dfrac{k^{2}-1}{12}$$
  • $$\dfrac{k^{2}-1}{6}$$
A multiple choice examination has $$5$$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get $$4$$ or more correct answers just by guessing is
  • $$\dfrac{13}{3^{5}}$$
  • $$\dfrac{11}{3^{5}}$$
  • $$\dfrac{10}{3^{5}}$$
  • $$\dfrac{17}{3^{5}}$$
Numbers are selected at random, one at a time, from the two digit numbers $$00, 01, 02 ..... 99$$ with replacement. An event $$E$$ occurs if and only if the product of two digits of a selected number is $$18$$. If four numbers are selected, then the probability that the event occurs atleast $$3$$ times is
  • $$\dfrac{96}{(25)^{4}}$$
  • $$\dfrac{97}{(25)^{4}}$$
  • $$\dfrac{95}{(25)^{4}}$$
  • $$\dfrac{94}{(28)^{4}}$$
If $$3$$% of electric bulbs manufactured by a company are defective, the probability that in a sample of $$100$$ bulbs exactly five are defective is
  • $$\displaystyle \frac{e^{-0.03}(0.03)^{5}}{ 5!}$$
  • $$\displaystyle \frac{e^{-0.03}(0.3)^{5}}{ 5!}$$
  • $$\displaystyle \frac{e^{-3}3^{5}}{ 5!}$$
  • $$\displaystyle \frac{e^{-0.03}3^{5}}{ 5!}$$
N coins, each with probability $$p$$ of getting head are tossed together.  In the options q = 1-p,  The probability of getting odd number of heads is
  • $$\displaystyle \frac{1-(q-p)^{n}}{2}$$
  • $$\displaystyle \frac{1+(q-p)^{n}}{2}$$
  • $$\displaystyle \frac{(q-p)^{n}}{2}$$
  • $$\displaystyle \frac{(q+p)^{n}}{2}$$
There are $$500$$ boxes each containing $$1000$$ ballot papers for election. The chance that a ballot paper is defective is $$0.002$$. Assuming that the number of defective ballot papers follow Poisson distribution, the number of boxes containing at least one defective ballot paper given that $$e^{-2}=0.1353$$ is
  • $$216$$
  • $$432$$
  • $$648$$
  • $$234$$
A telephone switch board receiving number of phone calls follows Poisson distribution with parameter $$3$$ per hour. The probability that it receives $$5$$ calls in $$2$$ hours duration is
  • $$\displaystyle \frac{e^{-3}3^{5}}{ 5!}$$
  • $$\displaystyle1-\frac{e^{-3}3^{5}}{ 5!}$$
  • $$\displaystyle \frac{e^{-6}3^{5}}{ 5!}$$
  • $$\displaystyle 1-\frac{e^{-6}3^{5}}{5!}$$
The incidence of an occupational disease to the workers of a factory is found to be $$\displaystyle \frac{1}{5000}$$ . If there are $$10,000$$ workers in a factory then the probability that none of them will get the disease is
  • $$e^{-1}$$
  • $$e^{-2}$$
  • $$e^{3}$$
  • $$e^{4}$$
One hundred identical coins each with probability P of showing up heads are tossed. If $$0<P<1$$ 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
  • $$\dfrac{50}{100}$$
  • $$\dfrac{51}{101}$$
  • $$\dfrac{52}{101}$$
  • $$\dfrac{53}{101}$$
The probability that atmost $$5$$ defective fuses will be found in a box of $$200$$ fuses, if experience shows that $$20 \%$$ of such fuses are defective,  is
  • $$\displaystyle \frac{e^{-40}40^{5}}{ 5!}$$
  • $$\displaystyle \sum_{x=0}^{5}\frac{e^{-40}40^{x}}{ x!}$$
  • $$\displaystyle \sum_{x=6}^{\infty}\frac{e^{-40}40^{x}}{ x!}$$
  • $$1-\displaystyle \sum_{x=6}^{\infty}\frac{e^{-40}40^{x}}{ x!}$$
Six unbiased coins are tossed $$6400$$ times. Using Poisson distribution, the approximate probability of getting six heads $$2$$ times is
  • $$\displaystyle \frac{e^{-64}(64)^{2}}{ 2!}$$
  • $$\displaystyle \frac{e^{-100}(100)^{2}}{ 2!}$$
  • $$1-\displaystyle \frac{e^{-100}(100)^{x}}{ x!}$$
  • $$\displaystyle \frac{e^{-100}(100)^{x}}{x!}$$
In a big city, $$5$$ accidents take place over a period of $$100$$ days. If the numebr of accidents follows P.D., the probability that there will be $$2$$ accidents in a day is
  • $$\displaystyle \frac{e^{-5}5^{2}}{ 2!}$$
  • $$\displaystyle \frac{e^{-05}5^{2}}{ 2!}$$
  • $$\displaystyle \frac{e^{-005}(0.05)^{2}}{ 2!}$$
  • $$\displaystyle \frac{e^{5}5^{2}}{ 2!}$$
If $$2$$% of pipes manufactured by a company are defective, the probability that in a sample of $$1000$$ pipes, exactly $$6$$ pipes are defective is (Assume Poisson distribution for the result)
  • $$\displaystyle \frac{e^{-2}2^{6}}{ 6!}$$
  • $$\displaystyle \frac{e^{-20}(20)^{6}}{ 6!}$$
  • $$e^{-20}$$
  • $$e^{-2}$$
A bag contains a very large number of white and black marbles in the ratio 1 :Two samples of marbles, 5 each are picked up. The probability that the first sample contains exactly one black and the second exactly three marbles is 
  • $$\dfrac{10}{3^{9}}$$
  • $$\dfrac{1600}{3^{10}}$$
  • $$\dfrac{800}{3^{10}}$$
  • $$\dfrac{10}{3^{5}}$$
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
  • $$12$$
  • $$11$$
  • $$10$$
  • $$13$$
A company knows on the basis of past experience that $$2$$% of the blades are defective. The probability of having 3 defective blades in a sample of $$100$$ blades is
  • $$e^{-2}2^{2}$$
  • $$\displaystyle \frac{e^{-2}2^{3}}{3!}$$
  • $$\displaystyle \frac{e^{-2}2^{3}}{ 2!}$$
  • $$\displaystyle \frac{e^{-4}2^{-1}}{ 2!}$$
A car hire firm has $$2$$ cars which it hires out day by day. If the number of demands for a car on each day follows poisson distribution with parameter $$1.5$$, then the probability that neither car is used is
  • $$e^{-1.5}$$
  • $$1.5\times e^{-1.5}$$
  • $$1-2.5\times e^{-1.5}$$
  • $$1-1.5\times e^{-1.5}$$
If $$X$$ and $$Y$$ are independent binomial variates $$B(5,1/2)$$ and $$B(7,1/2)$$, then find the value of $$P(X+Y=3)$$.
  • $$P(X+Y=3)=\cfrac{45}{1024}$$
  • $$P(X+Y=3)=\cfrac{65}{1024}$$
  • $$P(X+Y=3)=\cfrac{55}{1024}$$
  • $$P(X+Y=3)=\cfrac{75}{1024}$$
A die is thrown $$7$$ times. What is the chance that an odd number turns up exactly $$4$$ times?
  • $$\cfrac{1}{2}$$
  • $$\cfrac{35}{128}$$
  • $$\cfrac{37}{128}$$
  • $$\cfrac{63}{128}$$
A can take a step forward with probability $$0.4$$ and backward with probability $$0.6$$. find the probability that at the end of eleven steps he is one step away from the starting point.
  • probability$$=0.57$$
  • probability$$=0.63$$
  • probability$$=0.43$$
  • probability$$=0.37$$
In a series of n independent trials for an event of constant probability p, the most probable number r of successes is given by $$\left ( n+1 \right )p-1< r< \left ( n+1 \right )p$$. Hence, the most probable number of successes is the integral part of $$\left ( n+1 \right )p$$. But if $$\left ( n+1 \right )p$$ is an integer, the chance of r successes is equal to that of $$r+1$$ successes and both $$r,r+1$$ are most probable numbers of successes.       
A bag contains 2 white balls and 1 black ball. A ball is drawn at random and returned to the bag.
The experiment is done 10 times. The probability that a white ball is drawn exactly 5 times is 
  • $$\dfrac{10!}{5!5!}\left ( \dfrac{2}{3} \right )^5$$
  • $$\dfrac{10!}{5!5!}\left ( \dfrac{1}{3} \right )^5$$
  • $$\dfrac{10!}{\left(5! \right)^2}\left(\dfrac{2}{9} \right )^5$$
  • $$\dfrac{10!}{5!}\left(\dfrac{2}{9} \right )^5$$
If $$n$$ different apples are to be distributed among $$m$$ children, find the chance that a particular child recieves $$p$$ apples.
  • $$\cfrac { { _{ }^{ n }{ C } }_{ p }{ m }^{ n-p } }{ { m }^{ n } } $$
  • $$\cfrac { { _{ }^{ n }{ C } }_{ p }{ (m+1) }^{ n-p } }{ { m }^{ n } } $$
  • $$\cfrac { { _{ }^{ n }{ C } }_{ p }{ (m-1) }^{ n-p } }{ { m }^{ n } } $$
  • $$\cfrac { { _{ }^{ n }{ C } }_{ p }{ (m-2) }^{ n-p } }{ { m }^{ n } } $$
From a box containing $$20$$ tickets of value $$1$$ to $$20$$, four tickets are drawn one by one. After each draw, the ticket is replaced. The probability that the largest value of tickets drawn is $$15$$ is 
  • $$\displaystyle { \left( \frac { 3 }{ 4 } \right) }^{ 4 }$$
  • $$\displaystyle \frac { 27 }{ 320 } $$
  • $$\displaystyle \frac { 27 }{ 1280 } $$
  • none of these
Let $$A$$ be a set containing $$n$$ elements. A subset $$P$$ of the set $$A$$ is chosen at random.
The set $$A$$ is reconstructed by replacing the element of $$P$$, and another subsets $$Q$$ of $$A$$ is chosen at random. The probability that $$\left( P\cap Q \right)  $$ contains exactly $$m(m<n)$$ elements is
  • $$\displaystyle \frac { { 3 }^{ n-m } }{ { 4 }^{ n } } $$
  • $$\displaystyle \frac { _{ }^{ n }{ { C }_{ m } }{ 3 }^{ m } }{ { 4 }^{ n } } $$
  • $$\displaystyle \frac { _{ }^{ n }{ { C }_{ m } }{ 3 }^{ n-m } }{ { 4 }^{ n } } $$
  • none of these
An unbiased die with faces marked $$1,2,3,4,5$$ and $$6$$ is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than $$2$$ and the maximum face value is not greater than $$5$$ is
  • $$\displaystyle \frac { 16 }{ 81 } $$
  • $$\displaystyle \frac { 1 }{ 81 } $$
  • $$\displaystyle \frac { 80 }{ 81 } $$
  • $$\displaystyle \frac { 65 }{ 81 } $$
A die is thrown $$(2n+1)$$ times, $$n\in N$$. The probability that faces with even numbers show odd number of times is
  • $$\displaystyle \frac { 2n+1 }{ 2n+3 } $$
  • less that $$\displaystyle \frac { 1 }{ 2 } $$
  • greater than $$\displaystyle \frac { 1 }{ 2 } $$
  • None of these
Suppose the probability for A to win a game against B is 0.If A has an option of playing either "best of 3 games" or a "best of 5 games" match against B, which option should A choose so that the probability of his winning the match is higher ? (No game ends in draw)
  • best of 3 games
  • best of 5 games
  • same probability in both
  • cannot win
Suppose the probability for $$A$$ to win a game against $$B$$ is $$0.4$$. If $$A$$ has an option playing either a "best of $$3$$ games" or a "best of 5 games" match against $$B$$, which option should $$A$$ choose so that the probability of his winning the match is higher? (No game ends in a draw).
  • Best of $$5$$ games
  • Best of $$3$$ games
  • Either best of $$3$$ games or best of $$5$$ games
  • None of the above
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