Class 11 Mathematics Binomial Distribution MCQs Set 03

Question. A coin tossed n times. If the probability that 4, 5, 6 heads occur are in A.P., then n =
(a) 14
(b) 8
(c) 15
(d) 11
Answer: (a) 14

Question. Suppose X follows binomial distribution with parameters n = 100 and \( p = \frac{1}{8} \) then P(x=r) is maximum when r =
(a) 49
(b) 50
(c) 12
(d) 34
Answer: (c) 12

Question. The probability of a bomb hitting a bridge is \( \frac{1}{2} \) and two direct hits are needed to destroy it. The least number of bombs required so that the probability of the bridge being destroyed is greater than 0.9 is
(a) 5
(b) 6
(c) 8
(d) 7
Answer: (d) 7

Question. A coin whose faces are marked 3 & 5 is tossed 4 times. The probability that the sum of the numbers thrown is greater than 15
(a) \( \frac{11}{16} \)
(b) \( \frac{5}{16} \)
(c) \( \frac{5}{8} \)
(d) \( \frac{1}{16} \)
Answer: (a) \( \frac{11}{16} \)

Question. The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successess is
(a) 128/256
(b) 219/256
(c) 37/256
(d) 28/256
Answer: (d) 28/256

Question. If the mean and the variance of a binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than one is equal to
(a) \( \frac{1}{16} \)
(b) \( \frac{5}{16} \)
(c) \( \frac{11}{16} \)
(d) \( \frac{15}{16} \)
Answer: (c) \( \frac{11}{16} \)

Question. The mean and the variance of a random variable X having a binomial distribution are 4 and 2 respectively, then P(X = 1) is
(a) 1/32
(b) 1/16
(c) 1/8
(d) 1/4
Answer: (a) 1/32

Question. A card is drawn and replaced three times from an ordinary pack of 52 playing cards. The probability that three spades are drawn is
(a) 9/64
(b) 1/64
(c) 37/64
(d) 3/64
Answer: (b) 1/64

Question. If we take 1280 sets each of 10 tosses of a fair coin, the number of sets we expect to get 7 heads and 3 tails is
(a) 450
(b) 300
(c) 150
(d) 75
Answer: (c) 150

Question. 201 coins each with probability P(0 < p < 1) of showing head are tossed together. If the probability of getting 100 heads is equal to the probability of getting 101 heads, then the value of p is
(a) 1/4
(b) 1/3
(c) 1/2
(d) 1/6
Answer: (c) 1/2

Question. The probability of a man hitting the target is \( \frac{1}{4} \). The number of times atleast must he fire so that the probability of his hitting the target atleast once is greater than \( \frac{2}{3} \) is
(a) 6
(b) 5
(c) 4
(d) 3
Answer: (c) 4

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 atleast one success is
(a) 49/169
(b) 84/169
(c) 120/169
(d) 36/169
Answer: (c) 120/169

Question. One in 9 ships is likely to be wrecked, when they are set on sail. When 6 ships set on sail, the probability for exactly, 3 will not arrive safely is
(a) \( \frac{25 \times 8^3}{9^6} \)
(b) \( 1 - \frac{1}{9^6} \)
(c) \( ^6C_3 \left(\frac{8^3}{9^6}\right) \)
(d) \( ^6C_3 \left(\frac{8^6}{9^3}\right) \)
Answer: (c) \( ^6C_3 \left(\frac{8^3}{9^6}\right) \)

Question. The largest possible variance of a binomial variate is
(a) n
(b) n/2
(c) n/4
(d) n/6
Answer: (c) n/4

Question. Five coins whose faces are marked 2 and 3 are thrown. The chance of obtaining a total of 12 is
(a) 11/16
(b) 15/16
(c) 5/16
(d) 1/16
Answer: (c) 5/16

Question. An experiment succeeds twice as often as it fails . The chance that in the next six trials, there shall be atleast four successes is
(a) \( ^6C_4 \left(\frac{2}{3}\right)^4 \left(\frac{1}{3}\right)^2 \)
(b) \( \sum_{x=0}^{4} {}^6C_x \left(\frac{2}{3}\right)^x \left(\frac{1}{3}\right)^{6-x} \)
(c) \( \sum_{x=4}^{6} {}^6C_x \left(\frac{2}{3}\right)^x \left(\frac{1}{3}\right)^{6-x} \)
(d) \( \sum_{x=6}^{8} {}^6C_x \left(\frac{2}{3}\right)^x \left(\frac{1}{3}\right)^{6-x} \)
Answer: (c) \( \sum_{x=4}^{6} {}^6C_x \left(\frac{2}{3}\right)^x \left(\frac{1}{3}\right)^{6-x} \)

Question. The probability that a bulb produced by a factory will fuse after 100 days of use is 0.05. The probability that out of 5 such bulbs atleast one will fuse after 100 days of use is
(a) \( (0.95)^5 \)
(b) \( (0.05)^5 \)
(c) \( 1 - (0.95)^5 \)
(d) \( 1 - (0.05)^5 \)
Answer: (c) \( 1 - (0.95)^5 \)

Question. The probability that a bulb produced by a factory will fuse after 100 days of use is 0.05. The probability that out of 5 such bulbs none of them fuse after 100 days is
(a) \( (0.05)^5 \)
(b) \( (0.95)^5 \)
(c) \( 1 - (0.95)^5 \)
(d) \( 1 - (0.05)^5 \)
Answer: (b) \( (0.95)^5 \)

Question. A coin whose faces are marked 3 & 5 is tossed 4 times. The probability that the sum of the numbers thrown is 12 is
(a) 1/16
(b) 5/16
(c) 5/8
(d) 1/8
Answer: (a) 1/16

Question. The sum and product of mean and variance of a binomial distribution are 24 and 128 respectively. The binomial distribution is
(a) \( \left(\frac{1}{2} + \frac{1}{2}\right)^{32} \)
(b) \( \left(\frac{3}{10} + \frac{7}{10}\right)^{32} \)
(c) \( \left(\frac{1}{50} + \frac{49}{50}\right)^{32} \)
(d) \( \left(\frac{1}{3} + \frac{2}{3}\right)^{32} \)
Answer: (a) \( \left(\frac{1}{2} + \frac{1}{2}\right)^{32} \)

Question. X and Y are independent binomial variates \( B\left(5, \frac{1}{2}\right) \) and \( B\left(7, \frac{1}{2}\right) \) then P(X + Y = 3) is
(a) 45/1024
(b) 55/1024
(c) 65/1024
(d) 60/1024
Answer: (b) 55/1024

Question. A die is tossed twice. Getting ‘an odd number’ is termed a success. The probability distribution of number of successess (X) is formed. Then its mean, variance are
(a) 1, 1/2
(b) 1/2, 1
(c) 1/2, 1/2
(d) 1, 1
Answer: (a) 1, 1/2

Question. If X follows a binomial distribution with parameters n = 8 and \( P = \frac{1}{2} \) then \( P(|x - 4| \le 2) = \)
(a) 119/128
(b) 9/128
(c) 101/128
(d) 11/128
Answer: (a) 119/128

Question. Consider 5 independent Bernouli's trials each with probability of success p. If the probability of at least one failure is greater than or equal to \( \frac{31}{32} \), then p lies in the interval (AIE-2011)
(a) \( \left[0, \frac{1}{2}\right] \)
(b) \( \left(\frac{1}{2}, 1\right] \)
(c) \( \left(\frac{1}{2}, \frac{3}{4}\right] \)
(d) \( \left(\frac{3}{4}, \frac{11}{12}\right] \)
Answer: (a) \( \left[0, \frac{1}{2}\right] \)

Question. A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is (AIE-2007)
(a) 1/729
(b) 8/9
(c) 8/729
(d) 8/243
Answer: (d) 8/243

Question. If on an average 1 vessel in every 10 is wrecked, the chance that out of 5 vessels expected 4 at least will arrive safely is
(a) 4/5
(b) 1/2
(c) 2/5
(d) \( 14 \left(\frac{9^5}{10^5}\right) \)
Answer: (d) \( 14 \left(\frac{9^5}{10^5}\right) \)

Question. A man takes a step forward with probability 0.4 and backward with probability 0.6. The probability that at the end of eleven steps, he is just one step away from the starting point is
(a) 462(0.24)⁵
(b) 368(0.24)⁵
(c) 462(0.24)⁶
(d) 368(0.24)⁶
Answer: (a) 462(0.24)⁵

Question. A die is thrown 2n + 1 times. The probability of getting 1 or 3 or 5 atmost n times is
(a) 1/2
(b) 1/n
(c) \( \frac{n}{2n+1} \)
(d) \( \frac{1}{2n+1} \)
Answer: (a) 1/2

Question. 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
(a) \( \frac{11}{3^5} \)
(b) \( \frac{10}{3^5} \)
(c) \( \frac{17}{3^5} \)
(d) \( \frac{13}{3^5} \)
Answer: (a) \( \frac{11}{3^5} \)

Question. A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is
(a) 8/729
(b) 8/243
(c) 1/729
(d) 8/9
Answer: (b) 8/243

Question. Statement-I : A fair coin is tossed 100 times. The probability of getting tails an odd number of times is 1/2
Statement-II : A fair coin is tossed 99 times. The probability of getting tails an odd number of times is 1/2
Then which of the above statements are true.
(a) only I
(b) only II
(c) both I and II
(d) neither I nor II
Answer: (c) both I and II

Question. For the binomial distribution \( (q+p)^{n} = \left( \frac{1}{2} + \frac{1}{2} \right)^{10} \), if A = P(x=3), B = P(x = 6), C = P(x = 9) then the ascending order of A, B, C is
(a) C, A, B
(b) A, B, C
(c) A, C, B
(d) B, C, A
Answer: (a) C, A, B

Question. One hundred identical coins are thrown as each coin has the probability of getting head as p. Let x = number of coins showing heads then match the following conditions with p value.
A) \( P(x = 49) = P(x = 50) \)      1) \( \frac{1}{2} \)
B) \( P(x = 48) = P(x = 52) \)      2) \( \frac{51}{101} \)
C) \( P(x = r) = P(x = n - r) \)     3) \( \frac{50}{101} \)
                                                       4) \( \frac{57}{100} \)
The correct matching is
(a) A-2, B-3, C-1
(b) A-4, B-2, C-3
(c) A-1, B-2, C-3
(d) A-3, B-1, C-1
Answer: (d) A-3, B-1, C-1

Question. Observe the following statements
Assertion (A) : X is binomial variate with parameters 2n + 1 and p = \( \frac{1}{2} \) then P(x = odd values) = \( \frac{1}{2} \)
Reason (R) : If \( ^{n}C_{r} = C_{r} \) then \( C_{1} + C_{3} + C_{5} + \ldots = 2^{n-1} \). Then
(a) A is false, R is true
(b) A is true, R is true but R \( \nRightarrow \) A
(c) A is true, R is true and R \( \implies \) A
(d) A is true, R is false
Answer: (c) A is true, R is true and R
\( \implies \) A