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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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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers