CBSE Questions for Class 12 Commerce Applied Mathematics Standard Probability Distributions Quiz 10 - MCQExams.com

The minimum number of times a fair coin must be tossed so that the probability of getting at least one head is at least $$0.8$$, is
  • $$7$$
  • $$6$$
  • $$5$$
  • $$3$$
A fair coin is tossed $$n$$ times. Let $$X=$$ the number of times head occurs $$P(X=4),P(X=5)$$ and $$P(X=6)$$ are in A.P., then the value of $$n$$ can be
  • $$7$$
  • $$10$$
  • $$12$$
  • $$14$$
India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points $$0, 1$$ and $$2$$ are $$0.45, 0.05,$$ and $$0.50$$ respectively. Assuming that the outcomes are independent, the probability of India getting at least $$7$$ points is
  • $$0.8750$$
  • $$0.0875$$
  • $$0.0625$$
  • $$0.0250$$
$$A$$ is a set containing $$n$$ elements. A subset $$P$$ of $$A$$ is chosen at random. The set $$A$$ is reconstructed by replacing the elements of $$P$$. A subset $$Q$$ of $$A$$ is again chosen at random. Find the probability that $$P$$ and $$Q$$ have no common elements.
  • $${(3/4)}^{n}$$
  • $${(1/4)}^{n}$$
  • $${(2/3)}^{n}$$
  • $${(1/3)}^{n}$$
Numbers are selected at random, one at a time, from the two digit numbers 00, 01, 02......., 99 with replacement. An even E occurs if and if the product of the two digits of a selected number isIf four numbers are selected, find the probability that the event E occurs at least 3 times.
  • $$\displaystyle 97/(5)^4$$
  • $$\displaystyle 97/(25)^4$$
  • $$\displaystyle 98/(25)^4$$
  • $$\displaystyle 96/(25)^4$$
One hundred identical coins, each with probability, $$p$$, of showing up heads are tossed once. If $$0<p<1$$ and the probability of heads showing on fifty coins is equal to that of heads showing on $$51$$ coins, then the value of $$p$$ is :
  • $$1/2$$
  • $$49/101$$
  • $$50/101$$
  • $$51/101$$
A card is drawn from a pack. The card is replaced and the pack is reshuffled. If this is done six times, the probability that $$2$$ hearts, $$2$$ diamonds and $$2$$ black cards are drawn is
  • $$\displaystyle 90.\frac {1}{4}^6$$
  • $$\displaystyle \frac {45}{2}.\frac {3}{4}^4$$
  • $$\displaystyle \frac {90}{2^{10}}$$
  • none of these
Suppose $$X$$ follows a binomial distribution with parameters $$n$$ and $$p$$, where $$0<p<1$$. If $$\displaystyle \frac { P\left( X=r \right)  }{ P\left( X=n-r \right)  } $$ is independent of $$n$$ and $$r$$, then $$p$$ is equal to
  • $$\displaystyle \frac { 1 }{ 3 } $$
  • $$\displaystyle \frac { 1 }{ 2 } $$
  • $$\displaystyle \frac { 1 }{ 4 } $$
  • None of these
A coin is tossed $$n$$ times. The probability of getting at least one head is greater than that of getting at least two tails by $$\displaystyle \frac {5}{32}$$. Then $$n$$ is
  • $$5$$
  • $$10$$
  • $$15$$
  • none of these
A coin is tossed 10 times, probability that on the $$ 10^{th} $$ throw to observed $$ 5^{th} $$ head is
  • $$\displaystyle ^{9}\textrm{C}_{5}\:\displaystyle \frac{1}{2^{5}} $$
  • $$\displaystyle\frac{ ^{9}\textrm{C}_{4}}{2^{10}} $$
  • $$\displaystyle\frac{ ^{9}\textrm{C}_{5}}{2^{10}} $$
  • None of these
A dice is thrown $$2n + 1$$ times, $$\displaystyle n \: \epsilon \: N$$. The probability that faces with even numbers show odd number of times is
  • $$\displaystyle \frac {2n + 1}{4n + 3}$$
  • less than $$\displaystyle \frac {1}{2}$$
  • greater than $$\displaystyle \frac {1}{2}$$
  • none of these
There is $$30%$$ chance that it rains on any particular day. Given that there is at least one rainy day in a week, what is the probability that there are at least two rainy days in the week?
  • $$\displaystyle \frac { 1-4 (0.7) ^7}{ 1-(0.7)^7 } $$
  • $$\displaystyle \frac { 4 (0.7) ^7}{ 1-(0.7)^7 } $$
  • $$\displaystyle \frac { 1- (0.7) ^7}{ 1-4.(0.7)^7 } $$
  • $$\displaystyle \frac { (0.7) ^7}{ 1-4.(0.7)^7 } $$
A bag contains $$14$$ balls of two colours, the number of balls of each colour being the same. $$7$$ balls are drawn at random one by one. The ball in hand is returned to the bag before each new draw. If the probability that at least $$3$$ balls of each colour are drawn is $$p$$ then
  • $$\displaystyle p > \frac {1}{2}$$
  • $$\displaystyle p = \frac {1}{2}$$
  • $$\displaystyle p < 1$$
  • $$\displaystyle p < \frac {1}{2}$$
An ordinary dice is rolled a certain number of times. If the probability of getting an odd number $$2$$ times is equal to the probability of getting an even number $$3$$ times. Then the probability of getting an odd number for odd number of times is:
  • $$\displaystyle \frac {1}{32}$$
  • $$\displaystyle \frac {5}{16}$$
  • $$\displaystyle \frac {1}{2}$$
  • $$\dfrac{3}{16} $$
A factory $$A$$ produces  $$10\%$$  defective valves and another factory $$B$$ produces  $$20\%$$  defective. A bag contains  $$4$$  valves of factory $$A$$ and $$ 5$$  valves of factory $$B$$. If two valves are drawn at random from the bag. Find the probability that at least one valves is defective. Give your answer upto two places of decimals.
  • $$0.19$$
  • $$0.39$$
  • $$0.29$$
  • $$0.25$$
A fair coin is tossed $$99$$ times. If $$X$$ is the number of times heads occurs $$P(X=r)$$ is maximum when $$r$$, is
  • $$49$$
  • $$50$$
  • $$51$$
  • None of these
The sum and product of mean and variance of a Binomial distribution are $$24$$ and $$128$$ respectively then the value of $$n$$ is
  • $$16$$
  • $$32$$
  • $$24$$
  • None of these
  • Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
  • Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
  • Assertion is true but Reason is false
  • Assertion is false but Reason is true
A coin is tossed $$\left (2n + 1 \right )$$ times, the probability that head appear odd number of times is
  • $$\displaystyle \frac{n}{2n+1}$$
  • $$\displaystyle \frac{n+1}{2n+1}$$
  • $$\displaystyle \frac{1}{2}$$
  • None of these
If $$x$$ is a random poission variable such that $$ B= P\left ( X= 2 \right )= P\:\left ( X= 3 \right ) $$, then $$ P\left ( X= 4 \right ) $$ equals
  • $$ 2B $$
  • $$ \displaystyle \frac{3B}{4} $$
  • $$ \displaystyle \frac{3B^{2}}{4} $$
  • None of these
In a binomial distribution $$\displaystyle B\left(n,p=\frac{1}{4}\right)$$  if the probability of at least one success is greater than or equal to $$\displaystyle \frac{9}{10}$$, then $$n$$ is greater than
  • $$\displaystyle \frac{1}{\log_{10}4+\log_{10}3}$$
  • $$\displaystyle \frac{9}{\log_{10}4-\log_{10}3}$$
  • $$\displaystyle \frac{4}{\log_{10}4-\log_{10}3}$$
  • $$\displaystyle \frac{1}{\log_{10}4-\log_{10}3}$$
If $$3 percent $$ bulb manufactured by a company are defective; the probability that in a sample of $$100$$ bulbs exactly five defective is
  • $$\dfrac { { { e }^{ -0.003 } }\left( 0.03 \right) ^{ 5 } }{  5! }$$
  • $$\dfrac { { { e }^{ -0.3 } }0.03^{ 5 } }{ 5!}$$
  • $$\dfrac { { { e }^{ -3 } }3^{ 5 } }{5! }$$
  • $$\dfrac { { e }^{ -0.3 }{ 3 }^{ -5 } }{5! }$$
In $$n$$ independent trials (finite) of a random experiment, let $$X$$ be the number of times an event $$A$$ occurs. If the probability of success of one trial say $$p$$ and we get the probability of failure of event $$A$$ i.e. $$P\left ( \bar{A} \right )=1-p=q$$. The probability of $$r$$ success in $$n$$ trials is denoted by$$ P(X = r)$$ such that $$P(X = r) = ^{n}C_{r}p^{r}q^{n-r}$$, known as binomial distribution of random variable $$X$$. We also have the following result-
(i) Probability of getting at least $$k$$ successes is
$$P\left ( X\geq K \right )=\sum_{r=k}^{n}\ ^{n}C_{r}p^{r}q^{n-r}$$
(ii) Probability of getting at the most $$k$$ successes is
$$P\left ( X\leq  K \right )=\sum_{r=0}^{n}\ ^{n}C_{r}p^{r}q^{n-r}$$
(iii) $$\sum_{r=0}^{n}=^nC_{r}p^{r}q^{n-r}=\left ( p+q \right )^{n}=1$$

On the basis of above information answer the following question.

In a hurdle race, a player has to cross $$10$$ hurdles.The probability that he will cross each hurdle is $$\displaystyle \frac{5}{6}$$, then the probability that he will knock down fewer than two hurdle is
  • $$\displaystyle 7\left (\frac{5}{6} \right )^{10}$$
  • $$\displaystyle 5\left (\frac{5}{6} \right )^{10}$$
  • $$\displaystyle 3\left (\frac{5}{6} \right )^{10}$$
  • $$\displaystyle \frac{3}{5}\left (\frac{5}{6} \right )^{10}$$
The probability of a man hitting a target in one trial is $$\displaystyle \frac{1}{4}$$.The chances of hitting the target at least once in $$n$$ trials exceeds $$\displaystyle \frac{2}{3}$$, then the value of $$n$$ equals
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
A pair of fair dice is thrown independently three times. The probability of getting a score of 9 exactly twice is
  • $$8/729$$
  • $$8/243$$
  • $$1/729$$
  • $$8/9$$
The problem of points. Two player $$A$$ and $$B$$ want respectively $$m$$ and $$n$$ points of winning a single game are $$p$$ and $$q$$ respectively where $$p+q =1$$. The stake is to belong to the player who first makes up his set; find the probabilities in favour of each player.
  • $$\displaystyle P^{m}\left [ 1+mq+\frac{m(m+1)}{1.2} \right ]q^{2}+...\frac{(m+n-2)!}{(m-1!)(n-1)!}q^{n-1}$$,$$\displaystyle q^{n}\left [ 1+np+\frac{n(n+1)}{1.2}p^{2}+......+\frac{(m+n-2)}{(m-1)!(n-1)!}p^{m-1} \right ]$$
  • $$\displaystyle P^{m}\left [ 1+mq+\frac{m(m+1)}{1.2} \right ]q^{2}+...\frac{(m+n-1)!}{(m-1!)(n-1)!}q^{n-1}$$,$$\displaystyle q^{n}\left [ 1+np+\frac{n(n+1)}{1.2}p^{2}+......+\frac{(m+n-2)}{(m-1)!(n-1)!}p^{m-1} \right ]$$
  • $$\displaystyle P^{m}\left [ 1+mq+\frac{m(m+1)}{1.2} \right ]q^{2}+...\frac{(m+n-2)!}{(m-1!)(n-1)!}q^{n-1}$$,$$\displaystyle q^{n}\left [ 1+np+\frac{n(n+1)}{1.2}p^{2}+......+\frac{(m+n-1)}{(m-1)!(n-1)!}p^{m-1} \right ]$$
  • $$\displaystyle P^{m}\left [ 1+mq+\frac{m(m+1)}{1.2} \right ]q^{2}+...\frac{(m+n-1)!}{(m-1!)(n-1)!}q^{n-1}$$,$$\displaystyle q^{n}\left [ 1+np+\frac{n(n+1)}{1.2}p^{2}+......+\frac{(m+n-1)}{(m-1)!(n-1)!}p^{m-1} \right ]$$
One hundred identical coins, each with probability $$p$$, of showing up heads are tossed. If $$\displaystyle 0< p< 1$$ and the probability of heads showing on $$50$$ coins is equal to that of the heads showing in $$51$$ coins, then the value of $$p$$ is
  • $$\displaystyle \frac{1}{2}$$
  • $$\displaystyle \frac{49}{101}$$
  • $$\displaystyle \frac{50}{101}$$
  • $$\displaystyle \frac{51}{101}$$
 In a certain experiment the probability of success is twice the probability of faliure. Find the probaility of at least four successes in six trails.
  • $$\displaystyle \frac{297}{729}$$
  • $$\displaystyle \frac{233}{729}$$
  • $$\displaystyle \frac{432}{729}$$
  • $$\displaystyle \frac{496}{729}$$
The distribution of a random variable $$X$$ whose range is $$\{ 1,2,3,4 \}$$  is given in the following table. Find the mean and variance of $$X$$ respectively are:

$$ X$$$$ 1$$$$ 2$$$$ 3$$$$ 4$$
$$ P(X = x)$$$$ K$$$$ 2K$$$$ 3K$$$$ 4K$$
  • $$2,3$$
  • $$3,1$$
  • $$3,2$$
  • $$2,4$$
the sample has $$3$$ defective and $$5$$ non-defective bulbs.
  • $$\displaystyle \frac{4.89.87.86}{7.97.95.47.31}$$
  • $$\displaystyle \frac{5103}{12500000}$$
  • $$\displaystyle \frac{45927}{12500000}$$
  • $$\displaystyle \frac{413343}{25000000}$$
 $$A$$ and $$B$$ play a match in which the chances of their winning a game is $$\frac{1}{4}$$ for both of them and $$\frac{1}{2}$$ is the probability that match is being drawn. Match is finished as soon as either player wins two games. Find the probability that the match will be finished in $$4$$ or less games.
  • $$\dfrac{9}{16}$$
  • $$\dfrac{7}{16}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{4}$$
$$A$$ and $$B$$ play a match to be decided as soon as either has won two games. The chance of either winning a game is $$\displaystyle \dfrac{1}{20}$$ and of its being drawn $$\displaystyle \dfrac{9}{10}.$$ What is the the chance that the match is finished in $$10$$ or less games?
  • $$\displaystyle 0.18 $$ approx.
  • $$\displaystyle 0.16 $$ approx.
  • $$\displaystyle 0.17 $$ approx.
  • $$\displaystyle 0.15 $$ approx.
In five throws with a single die what is the chance of throwing (1) three aces exactly, (2) three aces at least?
  • $$\dfrac{125}{3888}$$,$$\dfrac{625}{648}$$, Favourable way for event $$E$$ when one number is selected are $$(2,9)$$
  • $$\dfrac{125}{3888}$$,$$\dfrac{23}{648}$$, Favourable way for event $$E$$ when one number is selected are $$(2,9)$$
  • $$\dfrac{3763}{3888}$$,$$\dfrac{23}{648}$$, Favourable way for event $$E$$ when one number is selected are $$(2,9)$$
  • $$\dfrac{3763}{3888}$$,$$\dfrac{625}{648}$$, Favourable way for event $$E$$ when one number is selected are $$(2,9)$$
One hundred identical coins, each with probability $$p$$ of showing heads are tossed once. If $$0 < p < 1$$ and the probability of heads showing on $$50$$ coins is equal to that of heads showing on $$51$$ coins, the value of $$p$$ is
  • $$1/2$$
  • $$51/101$$
  • $$49/101$$
  • $$3/101$$
Suppose the probability for $$A$$ to win a game against $$B$$ is $$0.4, A$$ has two options of playing matches against $$B$$, One 'the best of $$3$$ games' and the other 'the best of $$5$$ games'. Which of the options, $$A$$ should choose?(No game ends in a draw).
  • best of $$3$$ games
  • best of $$5$$ games
  • both options are equally probable
  • can't say
Three fair coins are tarred simultaneously. Let x denote the number of heads of experiment.
  • Write the sample space of experiment.
  • Write the rang set of x
  • contract the probability distribution of x
  • Hence, find mean variance and standard deviation (s.d) of x
A man takes a step forward with probability 0.4 and backward with probability 0.Find the probability that at the end of eleven steps he is one step away from the starting point.
  • $$\displaystyle 0.368$$
  • $$\displaystyle 0.147$$
  • $$\displaystyle 0.22$$
  • $$\displaystyle 0.073$$
The probability that an event $$A$$ occurs in a single trial of an experiment is $$0.6$$. Three independent trials of the experiment are performed. The probability that the event $$A$$ occurs at least twice is
  • $$0.636$$
  • $$0.632$$
  • $$0.648$$
  • $$0.946$$
Each of two persons $$ A$$ and $$B$$ toss three fair coins. The probability that both get the same number of heads is
  • $$\dfrac {3}{8}$$
  • $$\dfrac {1}{9}$$
  • $$\dfrac {5}{16}$$
  • $$\dfrac {7}{16}$$
A fair coin is tossed $$100$$ times. The probability of getting tails $$1, 3, ..., 49$$ times is
  • $$1/2$$
  • $$1/4$$
  • $$1/8$$
  • $$1/16$$
In a hurdle race, a runner has probability $$p$$ of jumping over a specific hurdle. Given that in $$5$$ trials, the rummer succeeded $$3$$ times, the conditional probability that the runner had succeeded in the first trial is
  • $$\dfrac {3}{5}$$
  • $$\dfrac {2}{5}$$
  • $$\dfrac {1}{5}$$
  • $$\dfrac {4}{5}$$
The minimum number of times a fair coin must be tossed so that the probability of getting at least one head is at least $$0.8$$ is
  • $$7$$
  • $$6$$
  • $$5$$
  • $$3$$
The probability that $$X$$ wins the match after $$(n+1)$$ games, $$(n\geq 1)$$ is
  • $$na^2b^{n-1}$$
  • $$a^2(nb^{n-1}+n(n-1)b^{n-2}c)$$
  • $$na^2bc^{n-1}$$
  • none of these
A radar unit is used to measure speeds of cars on a motorway. The speeds are normally distributed with a mean of $$9$$ km/hr and a standard deviation of $$10$$ km/hr. What is the probability that a car picked at random is travelling at more than $$100$$ km/hr?
  • $$0.1698$$
  • $$0.1548$$
  • $$0.1587$$
  • $$0.1236$$
If $$X$$ follows a binomial distribution with parameter $$n=101$$ and $$p=1/3$$, then $$P(X=r)$$ is maximum if $$r$$ equals
  • $$34$$
  • $$33$$
  • $$32$$
  • $$35$$
If $$X$$ and $$Y$$ are the independent random variable for $$\displaystyle A\left( 5,\frac { 1 }{ 2 }  \right) $$ and $$\displaystyle B\left( 7,\frac { 1 }{ 2 }  \right) $$, then $$P\left( X+Y\ge 1 \right) =$$
  • $$\displaystyle \frac { 4095 }{ 4096 } $$
  • $$\displaystyle \frac { 309 }{ 4096 } $$
  • $$\displaystyle \frac { 4031 }{ 4096 } $$
  • none of these
Let, a binomial distribution $$\displaystyle B\left ( n,p=\frac{1}{4} \right )$$. If the probability of at least one success is greater than or equal to $$\displaystyle \frac{9}{10}$$, then $$n$$ is greater than
  • $$\displaystyle \frac{1}{\log _{10}4+\log _{10}3}$$
  • $$\displaystyle \frac{9}{\log _{10}4-\log _{10}3}$$
  • $$\displaystyle \frac{4}{\log _{10}4-\log _{10}3}$$
  • $$\displaystyle \frac{1}{\log _{10}4-\log _{10}3}$$
A fair coin is tossed $$100$$ times. The probability of getting tails an odd number of times is
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{4}$$
  • $$\dfrac {1}{8}$$
  • $$\dfrac {3}{8}$$
An experiment succeeds twice as often as it fails. Find the probability that in next 6 trials, there will be more than 3 successes
  • $$\displaystyle \frac {496}{729}$$
  • $$\displaystyle \frac {31}{729}$$
  • $$\displaystyle \frac {16}{27}$$
  • $$\displaystyle \frac {31}{54}$$
If the mean and variance of a binomial variate $$X$$ are $$8$$ and $$4$$ respectively, then $$P(X < 3) = $$
  • $$\dfrac{265}{2^{15}}$$
  • $$\dfrac{137}{2^{16}}$$
  • $$\dfrac{267}{2^{16}}$$
  • $$\dfrac{265}{2^{16}}$$
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