Class 11 Mathematics Binomial Distribution MCQs Set 02

Question. The mean of binomial distribution is 6 and its S.D. is \( \sqrt{2} \), then the number of trials n is
(a) 7
(b) 8
(c) 9
(d) 10
Answer: (c) 9

Question. If a random variable X follows B.D. with mean 2.4 and variance 1.44, the number of independent trials n is
(a) 10
(b) 8
(c) 6
(d) 2
Answer: (c) 6

Question. If for a binomial distribution with n = 5, \( 4p(X=1)=P(X=2) \), the probability of success is
(a) \( \frac{1}{3} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{1}{8} \)
Answer: (b) \( \frac{2}{3} \)

Question. A symmetrical die is rolled 720 times. Getting a face with four points is considered to be a success. The mean and variance of the number of successes is
(a) 20, 120
(b) 120, 100
(c) 100, 100
(d) 50, 50
Answer: (b) 120, 100

Question. The probability of getting atleast two heads when an unbiased coin is tossed three times is
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{1}{8} \)
Answer: (c) \( \frac{1}{2} \)

Question. For a binomial distribution \( \bar{x} = 4, \sigma^2 = 3 \) then the distribution of x is
(a) \( \left(\frac{1}{4} + \frac{3}{4}\right)^{16} \)
(b) \( \left(\frac{3}{4} + \frac{1}{4}\right)^{16} \)
(c) \( \left(\frac{1}{2} + \frac{1}{2}\right)^{16} \)
(d) \( \left(\frac{1}{2} + \frac{1}{2}\right)^8 \)
Answer: (b) \( \left(\frac{3}{4} + \frac{1}{4}\right)^{16} \)

Question. If for a binomial distribution mean = \( \frac{10}{3} \) and sum of mean and variance is \( \frac{40}{9} \). The parameters are
(a) \( \frac{2}{3}, 10 \)
(b) \( \frac{2}{3}, 20 \)
(c) \( 5, \frac{2}{3} \)
(d) \( 4, \frac{2}{3} \)
Answer: (c) \( 5, \frac{2}{3} \)

Question. If for a binomial distribution \( \bar{x} = \frac{6}{5} \) and the difference between mean and variance is \( \frac{6}{25} \). The number of trials is
(a) 8
(b) 7
(c) 6
(d) 5
Answer: (c) 6

Question. If for a B.D. with n=12, the ratio of variance to mean is \( \frac{1}{3} \), the probability of 10 successes is
(a) \( ^{15}C_{10} \left(\frac{2}{3}\right)^{10} \left(\frac{1}{3}\right)^2 \)
(b) \( ^{12}C_{10} \left(\frac{2}{3}\right)^{10} \left(\frac{1}{3}\right)^2 \)
(c) \( \left(\frac{2}{3}\right)^{10} \)
(d) \( \left(\frac{1}{3}\right)^{10} \)
Answer: (b) \( ^{12}C_{10} \left(\frac{2}{3}\right)^{10} \left(\frac{1}{3}\right)^2 \)

Question. A symmetrical die is thrown three times. If getting a six is considered to be a success, the probability of atleast two successes is
(a) \( \frac{4}{27} \)
(b) \( \frac{3}{27} \)
(c) \( \frac{2}{27} \)
(d) \( \frac{1}{27} \)
Answer: (c) \( \frac{2}{27} \)

Question. If for a BD the mean is 6 and standard deviation is \( \frac{1}{\sqrt{2}} \), the probability of success is
(a) 11/12
(b) 10/12
(c) 9/12
(d) 8/12
Answer: (a) 11/12

Question. If the sum of mean and variance of B.D. for 5 trials is 1.8, the binomial distribution is
(a) \( (0.8 + 0.2)^5 \)
(b) \( (0.2 + 0.8)^5 \)
(c) \( (0.8 + 0.2)^{10} \)
(d) \( (0.2 + 0.8)^{10} \)
Answer: (a) \( (0.8 + 0.2)^5 \)

Question. If for a binomial distribution n = 4 and 6P(X=4)=P(X=2), the probability of success is
(a) 3/4
(b) 1/2
(c) 1/3
(d) 1/4
Answer: (b) 1/2

Question. A box contains 6 red and 4 white marbles. A marble is drawn and replaced three times from the box. The probability that one white marble is drawn
(a) \( \frac{53}{125} \)
(b) \( \frac{54}{125} \)
(c) \( \frac{56}{125} \)
(d) \( \frac{52}{125} \)
Answer: (b) \( \frac{54}{125} \)

Question. A random variable X is binomially distributed with mean is 12 and variance is 8. Find the parameters of the distribution are
(a) \( 18, \frac{1}{3} \)
(b) \( 36, \frac{1}{3} \)
(c) \( 36, \frac{2}{3} \)
(d) \( 18, \frac{2}{3} \)
Answer: (b) \( 36, \frac{1}{3} \)

Question. If the mean and variance of a binomial variate X are 8 and 4 respectively then \( P(X < 3) = \) (EAM-2014)
(a) \( \frac{137}{2^{16}} \)
(b) \( \frac{697}{2^{16}} \)
(c) \( \frac{265}{2^{16}} \)
(d) \( \frac{265}{2^{15}} \)
Answer: (a) \( \frac{137}{2^{16}} \)

Question. The mean and standard deviation of a binomial variate X are 4 and \( \sqrt{3} \) respectively. Then \( P(X \ge 1) = \) (EAM-2007)
(a) \( 1 - \left(\frac{1}{4}\right)^{16} \)
(b) \( 1 - \left(\frac{3}{4}\right)^{16} \)
(c) \( 1 - \left(\frac{2}{3}\right)^{16} \)
(d) \( 1 - \left(\frac{1}{3}\right)^{16} \)
Answer: (b) \( 1 - \left(\frac{3}{4}\right)^{16} \)

Question. If the mean and variance of a binomial variable X are 2 and 1 respectively, then \( P(X \ge 1) = \) (EAM-2010)
(a) \( \frac{2}{3} \)
(b) \( \frac{15}{16} \)
(c) \( \frac{7}{8} \)
(d) \( \frac{4}{5} \)
Answer: (b) \( \frac{15}{16} \)

Question. X follows a binomial distribution with parameters \( n=6 \) and \( p \). If \( 4P(X=4)=P(X=2) \). Then \( p = \) (EAM-2009)
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{4} \)
(c) \( \frac{1}{6} \)
(d) \( \frac{1}{3} \)
Answer: (d) \( \frac{1}{3} \)

Question. Suppose X follows a binomial distribution with parameters n and p, where 0 < p < 1. If \( \frac{P(X=r)}{P(X=n-r)} \) is independent of n for every r, then p = (EAM-2012)
(a) 1/2
(b) 1/3
(c) 1/4
(d) 1/8
Answer: (a) 1/2

Question. For a binomial distribution if \( p = \frac{1}{4}, n = 20 \) the probability of mode is
(a) \( ^{20}C_{5} \left(\frac{3}{4}\right)^{5} \)
(b) \( ^{20}C_{5} \left(\frac{1}{4}\right)^{5} \left(\frac{3}{4}\right)^{15} \)
(c) \( ^{10}C_{10} \left(\frac{1}{4}\right)^{10} \left(\frac{3}{4}\right)^{10} \)
(d) \( ^{10}C_{10} \left(\frac{3}{4}\right)^{10} \)
Answer: (b) \( ^{20}C_{5} \left(\frac{1}{4}\right)^{5} \left(\frac{3}{4}\right)^{15} \)

Question. In a B.D. \( n = 400, p = \frac{1}{5} \). Its standard deviation is
(a) \( 10 \times \sqrt{2} \)
(b) \( \frac{1}{800} \)
(c) 4
(d) 8
Answer: (d) 8

Question. If X be B.V. with \( E(X) = 5 \) and \( E(X^{2}) - \{E(X)\}^{2} = 4 \), then the parameters of distribution are
(a) \( \frac{1}{4}, 20 \)
(b) \( \frac{1}{5}, 20 \)
(c) \( \frac{1}{5}, 25 \)
(d) \( \frac{4}{5}, 25 \)
Answer: (c) \( \frac{1}{5}, 25 \)

Question. If x is \( B\left(n, \frac{1}{3}\right), P(x \geq 1) > 0.8 \), the least value of n is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (b) 4

Question. A box contains 'a' white and 'b' black balls. 'c' balls are drawn at random with replacement. The expected number of white balls drawn is
(a) \( \frac{a}{a+b} \)
(b) \( \frac{ac}{a+b} \)
(c) \( \frac{bc}{a+b} \)
(d) \( \frac{b}{a+b} \)
Answer: (b) \( \frac{ac}{a+b} \)

Question. 5 cards are drawn one after another successively with replacement from a well shuffled pack of 52 cards. The probability that all the 5 cards are spades
(a) \( \left(\frac{3}{4}\right)^{5} \)
(b) \( 1 - \left(\frac{3}{4}\right)^{5} \)
(c) \( \left(\frac{1}{4}\right)^{5} \)
(d) \( 1 - \left(\frac{1}{4}\right)^{5} \)
Answer: (c) \( \left(\frac{1}{4}\right)^{5} \)

Question. A card is drawn and replaced four times from an ordinary pack of 52 playing cards. The probability that at least once heart is drawn
(a) \( \left(\frac{3}{4}\right)^{4} \)
(b) \( 1 - \left(\frac{1}{2}\right)^{4} \)
(c) \( 1 - \left(\frac{3}{4}\right)^{4} \)
(d) \( \left(\frac{1}{2}\right)^{4} \)
Answer: (c) \( 1 - \left(\frac{3}{4}\right)^{4} \)

Question. Out of 800 families with 4 children each the expected number of families having 2 boys and 2 girls is
(a) 100
(b) 200
(c) 300
(d) 400
Answer: (c) 300

Question. One hundred identical coins each with probability P of showing up heads are tossed. If O<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) \( \frac{50}{100} \)
(b) \( \frac{51}{101} \)
(c) \( \frac{52}{101} \)
(d) \( \frac{53}{101} \)
Answer: (b) \( \frac{51}{101} \)

Question. 2K+1 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
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{1}{8} \)
Answer: (c) \( \frac{1}{2} \)

Question. A card is drawn and replaced in an ordinary pack of playing cards. The number of times a card must be drawn so that the probability of getting atleast a club card is greater than \( \frac{3}{4} \)
(a) 7
(b) 6
(c) 5
(d) 4
Answer: (c) 5

Question. Out of 2000 families with 4 children, the number of familes you expect to have atleast one boy is
(a) 1875
(b) 750
(c) 1250
(d) 625
Answer: (a) 1875

Question. 6 symmetrical dice are thrown 1458 times. The number of times you expect three dice to show a four or five is
(a) 160
(b) 320
(c) 480
(d) 600
Answer: (b) 320

Question. If a sex ratio of births is 49 girls to 51 boys, the probability that there will be 8 girls amongst 10 babies born on the same day in a maternity hospital is
(a) \( ^{10}C_{8} (0.51)^{8} (0.49)^{2} \)
(b) \( ^{10}C_{8} (0.49)^{8} (0.51)^{2} \)
(c) \( ^{10}C_{8} (0.49 \times 0.51)^{8} \)
(d) \( ^{10}C_{8} (0.49 \times 0.51)^{2} \)
Answer: (b) \( ^{10}C_{8} (0.49)^{8} (0.51)^{2} \)

Question. In a market region half of the households is known to use a particular brand of soap. In a household survey, a sample of 10 house holds are alloted to each investigator and 2048 investigators are appointed for the survey. The number of investigators likely to report that there are three households is
(a) 240
(b) 352
(c) 1696
(d) 120
Answer: (a) 240

Question. In a market region half of the households is known to use a particular brand of soap. In a household survey, a sample of 10 house holds are alloted to each investigator and 2048 investigators are appointed for the survey. The number of investigators likely to report that there are atleast 4 users is
(a) 240
(b) 352
(c) 1696
(d) 120
Answer: (c) 1696

Question. The least number of times a fair coin is to be tossed in order that the probability of getting atleast one head is at least 0.99 is
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (c) 7

Question. An arcade game is such that the probability of any person winning is always 0.3. The minimum number of people play the game to ensure that the probability that atleast one person wins is greater than or equal to 0.96 is
(a) 8
(b) 9
(c) 10
(d) 12
Answer: (c) 10

Question. Suppose A and B are two equally strong table tennis players. Which of the following two events is more probable
a) A beats B in exactly 3 games out of 4
b) A beats B in exactly 5 games out of 8
(a) a
(b) b
(c) a & b
(d) neither a nor b
Answer: (a) a

Question. A bag contains 13 balls numbered from 1 to 13. Suppose drawing of an even number is a success. Two balls are drawn with replacement from the bag. The probability of getting two successes is
(a) \( \frac{84}{169} \)
(b) \( \frac{49}{169} \)
(c) \( \frac{36}{169} \)
(d) \( \frac{120}{169} \)
Answer: (c) \( \frac{36}{169} \)