Mathematics · Textbook solutions

Binomial Distribution

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 71 questions

8.1 Bernoulli Trial

1 q

Solved Example

Worked · 1
  1. 8.1.1 SolvedEx.1
    Six balls are drawn successively from an urn containing 7 red and 9 black balls. Tell whether or not the trials of drawing balls are Bernoulli trials when after each draw the ball drawn is (i) replaced (ii) not replaced in the urn.

8.2 Binomial Distribution

2 q

Solved Examples

Worked · 2
  1. 8.2 SolvedEx.1
    If a fair coin is tossed 10 times, find the probability of getting (i) exactly six heads (ii) at least six heads (iii) at most six heads
  2. 8.2 SolvedEx.2
    Ten eggs are drawn successively with replacement from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.

8.3 Mean and Variance of Binomial Distribution

3 q

Solved Examples

Worked · 3
  1. 8.3 ForExample.1
    If X∼B(10,0.4)X \sim B(10, 0.4) then find E(X)E(X) and Var(X)\text{Var}(X).
  2. 8.3 SolvedEx.1
    Let the p.m.f. of r.v. XX be P(X=x)=4Cx(59)x×(49)4−xP(X = x) = {}^{4}C_{x}\left(\dfrac{5}{9}\right)^{x} \times \left(\dfrac{4}{9}\right)^{4-x}, for x=0,1,2,3,4x = 0, 1, 2, 3, 4. Then find E(X)E(X) and Var(X)\text{Var}(X).
  3. 8.3 SolvedEx.2
    If E(X)=6E(X) = 6 and Var(X)=4.2\text{Var}(X) = 4.2, find nn and pp.

Exercise 8.1

27 q
  1. Ex 8.1 Q.1 (i)
    A die is thrown 6 times. If 'getting an odd number' is a success, find the probability of (i) 5 successes.
  2. Ex 8.1 Q.1 (ii)
    A die is thrown 6 times. If 'getting an odd number' is a success, find the probability of (ii) at least 5 successes.
  3. Ex 8.1 Q.1 (iii)
    A die is thrown 6 times. If 'getting an odd number' is a success, find the probability of (iii) at most 5 successes.
  4. Ex 8.1 Q.2
    A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.
  5. Ex 8.1 Q.3
    There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?
  6. Ex 8.1 Q.4 (i)
    Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. find the probability that (i) all the five cards are spades.
  7. Ex 8.1 Q.4 (ii)
    Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. find the probability that (ii) only 3 cards are spades.
  8. Ex 8.1 Q.4 (iii)
    Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. find the probability that (iii) none is a spade.
  9. Ex 8.1 Q.5 (i)
    The probability that a bulb produced by a factory will fuse after 150 days of use is 0⋅050 \cdot 05. Find the probability that out of 5 such bulbs (i) none will fuse after 150 days of use.
  10. Ex 8.1 Q.5 (ii)
    The probability that a bulb produced by a factory will fuse after 150 days of use is 0⋅050 \cdot 05. Find the probability that out of 5 such bulbs (ii) not more than one will fuse after 150 days of use.
  11. Ex 8.1 Q.5 (iii)
    The probability that a bulb produced by a factory will fuse after 150 days of use is 0⋅050 \cdot 05. Find the probability that out of 5 such bulbs (iii) more than one will fuse after 150 days of use.
  12. Ex 8.1 Q.5 (iv)
    The probability that a bulb produced by a factory will fuse after 150 days of use is 0⋅050 \cdot 05. Find the probability that out of 5 such bulbs (iv) at least one will fuse after 150 days of use.
  13. Ex 8.1 Q.6
    A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
  14. Ex 8.1 Q.7
    On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
  15. Ex 8.1 Q.8 (i)
    A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1/1001/100. find the probability that he will win a prize (i) at least once.
  16. Ex 8.1 Q.8 (ii)
    A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1/1001/100. find the probability that he will win a prize (ii) exactly once.
  17. Ex 8.1 Q.8 (iii)
    A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1/1001/100. find the probability that he will win a prize (iii) at least twice.
  18. Ex 8.1 Q.9 (i)
    In a box of floppy discs it is known that 95% will work. A sample of three of the discs is selected at random. Find the probability that (i) none of the sample will work.
  19. Ex 8.1 Q.9 (ii)
    In a box of floppy discs it is known that 95% will work. A sample of three of the discs is selected at random. Find the probability that (ii) 1 of the sample will work.
  20. Ex 8.1 Q.9 (iii)
    In a box of floppy discs it is known that 95% will work. A sample of three of the discs is selected at random. Find the probability that (iii) 2 of the sample will work.
  21. Ex 8.1 Q.9 (iv)
    In a box of floppy discs it is known that 95% will work. A sample of three of the discs is selected at random. Find the probability that (iv) all 3 of the sample will work.
  22. Ex 8.1 Q.10
    Find the probability of throwing at most 2 sixes in 6 throws of a single die.
  23. Ex 8.1 Q.11
    It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?
  24. Ex 8.1 Q.12 (i)
    Given that X∼B(n,p)X \sim B(n, p) (i) If n=10n = 10 and p=0⋅4p = 0 \cdot 4, find E(X)E(X) and Var(X)Var(X).
  25. Ex 8.1 Q.12 (ii)
    Given that X∼B(n,p)X \sim B(n, p) (ii) If p=0⋅6p = 0 \cdot 6 and E(X)=6E(X) = 6, find nn and Var(X)Var(X).
  26. Ex 8.1 Q.12 (iii)
    Given that X∼B(n,p)X \sim B(n, p) (iii) If n=25n = 25, E(X)=10E(X) = 10 find pp and SD(X)SD(X).
  27. Ex 8.1 Q.12 (iv)
    Given that X∼B(n,p)X \sim B(n, p) (iv) If n=10n = 10, E(X)=8E(X) = 8, find Var(X)Var(X).

Miscellaneous Exercise 8

38 q

Choose the correct option

Practice · 7
  1. Misc 8 I (1)
    A die is thrown 100 times. If getting an even number is considered a sucess, then the standard deviation of the number of successes is
    1. A.
      50\sqrt{50}
    2. B.
      55
    3. C.
      2525
    4. D.
      1010
  2. Misc 8 I (2)
    The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probablity of 2 successes is
    1. A.
      128256\dfrac{128}{256}
    2. B.
      219256\dfrac{219}{256}
    3. C.
      37256\dfrac{37}{256}
    4. D.
      28256\dfrac{28}{256}
  3. Misc 8 I (3)
    For a binomial distribution, n=5n = 5. If P(X=4)=P(X=3)P(X = 4) = P(X = 3) then p=…p = \ldots
    1. A.
      13\dfrac{1}{3}
    2. B.
      34\dfrac{3}{4}
    3. C.
      11
    4. D.
      23\dfrac{2}{3}
  4. Misc 8 I (4)
    In a binomial distribution, n=4n = 4. If 2P(X=3)=3P(X=2)2 P(X = 3) = 3 P(X = 2) then p=…p = \ldots
    1. A.
      413\dfrac{4}{13}
    2. B.
      513\dfrac{5}{13}
    3. C.
      913\dfrac{9}{13}
    4. D.
      613\dfrac{6}{13}
  5. Misc 8 I (5)
    If X∼B(4,p)X \sim B(4, p) and P(X=0)=1681P(X = 0) = \dfrac{16}{81}, then P(X=4)=…P(X = 4) = \ldots
    1. A.
      116\dfrac{1}{16}
    2. B.
      181\dfrac{1}{81}
    3. C.
      127\dfrac{1}{27}
    4. D.
      18\dfrac{1}{8}
  6. Misc 8 I (6)
    The probability of a shooter hitting a target is 34\dfrac{3}{4}. How many minimum number of times must he fire so that the probability of hitting the target at least once is more than 0⋅990 \cdot 99?
    1. A.
      22
    2. B.
      33
    3. C.
      44
    4. D.
      55
  7. Misc 8 I (7)
    If the mean and variance of a binomial distribution are 18 and 12 respectively, then n=…n = \ldots
    1. A.
      3636
    2. B.
      5454
    3. C.
      1818
    4. D.
      2727

Solve the following

Practice · 31
  1. Misc 8 Q.1 (i)
    Let X∼B(10,0⋅2)X \sim B(10, 0 \cdot 2), Find (i) P(X=1)P(X = 1).
  2. Misc 8 Q.1 (ii)
    Let X∼B(10,0⋅2)X \sim B(10, 0 \cdot 2), Find (ii) P(X≥1)P(X \geq 1).
  3. Misc 8 Q.1 (iii)
    Let X∼B(10,0⋅2)X \sim B(10, 0 \cdot 2), Find (iii) P(X≤8)P(X \leq 8).
  4. Misc 8 Q.2 (i)
    Let X∼B(n,p)X \sim B(n, p) (i) If n=10n = 10, E(X)=5E(X) = 5, find pp and Var(X)Var(X).
  5. Misc 8 Q.2 (ii)
    Let X∼B(n,p)X \sim B(n, p) (ii) If E(X)=5E(X) = 5 and Var(X)=2⋅5Var(X) = 2 \cdot 5, find nn and pp.
  6. Misc 8 Q.3 (i)
    If fair coin is tossed 10 times find the probability that it shows heads (i) 5 times.
  7. Misc 8 Q.3 (ii)
    If fair coin is tossed 10 times find the probability that it shows heads (ii) in the first four tosses and tail in last six tosses.
  8. Misc 8 Q.4
    Probability that bomb will hit target is 0⋅80 \cdot 8. Find the probability that out of 10 bombs dropped exactly 2 will miss the target.
  9. Misc 8 Q.5 (i)
    The probability that a mountain-bike rider travelling along a certain track will have a tyre burst is 0⋅050 \cdot 05. Find the probability that among 17 riders : (i) exactly one has a burst tyre.
  10. Misc 8 Q.5 (ii)
    The probability that a mountain-bike rider travelling along a certain track will have a tyre burst is 0⋅050 \cdot 05. Find the probability that among 17 riders : (ii) at most three have a burst tyre.
  11. Misc 8 Q.5 (iii)
    The probability that a mountain-bike rider travelling along a certain track will have a tyre burst is 0⋅050 \cdot 05. Find the probability that among 17 riders : (iii) two or more have burst tyres.
  12. Misc 8 Q.6
    Probability that a lamp in a classroom will burnt out will be 0⋅30 \cdot 3. Six lamps are fitted in the classroom. If it is known that the classroom is unusable if the number of lamps burning in it is less than four, find the probability that classroom can not used at random occasion.
  13. Misc 8 Q.7
    Lot of 100 items contains 10 defective items. Five items are selected at random from the lot and sent to the retail store. What is the probability that the store will receive at most one defective item?
  14. Misc 8 Q.8
    A large chain retailer purchases certain kind of electric device from manufacturer. The manufacturer indicates that the defective rate of the device is 3%. The inspector of the retailer picks 20 items from a shipment. What is the probability that the store will receive at most one defective item?
  15. Misc 8 Q.9
    The probability that the certain kind of component will survive a check test is 0⋅60 \cdot 6. Find the probability that exactly 2 of the next 4 components tested survive.
  16. Misc 8 Q.10
    An examination consists of 10 multiple-choice questions, in each of which a candidate has to deduce which one of five suggested answers is correct. A completely unprepared student guesses each answer completely randomly. What is the probability that this student gets 8 or more questions correct? Draw the appropriate moral !
  17. Misc 8 Q.11 (i)
    The probability that a machine will produce all bolts in a production run within specification is 0⋅9980 \cdot 998. A sample of 8 machines is taken at random. Calculate the probability that (i) all 8 machines will produce all bolts within specification.
  18. Misc 8 Q.11 (ii)
    The probability that a machine will produce all bolts in a production run within specification is 0⋅9980 \cdot 998. A sample of 8 machines is taken at random. Calculate the probability that (ii) 7 or 8 machines will produce all bolts within specification.
  19. Misc 8 Q.11 (iii)
    The probability that a machine will produce all bolts in a production run within specification is 0⋅9980 \cdot 998. A sample of 8 machines is taken at random. Calculate the probability that (iii) at least 6 machines will produce all bolts within specification.
  20. Misc 8 Q.12
    The probability that a machine develops a fault within the first 3 years of use is 0⋅0030 \cdot 003. If 40 machines are selected at random, calculate the probability that 38 or more will not develop any faults within the first 3 years of use.
  21. Misc 8 Q.13 (i)
    A computer installation has 10 terminals. Independently, the probability that any one terminal will require attention during a week is 0⋅10 \cdot 1. Find the probabilities that (i) 0 terminals will require attention during the next week.
  22. Misc 8 Q.13 (ii)
    A computer installation has 10 terminals. Independently, the probability that any one terminal will require attention during a week is 0⋅10 \cdot 1. Find the probabilities that (ii) 1 terminal will require attention during the next week.
  23. Misc 8 Q.13 (iii)
    A computer installation has 10 terminals. Independently, the probability that any one terminal will require attention during a week is 0⋅10 \cdot 1. Find the probabilities that (iii) 2 terminals will require attention during the next week.
  24. Misc 8 Q.13 (iv)
    A computer installation has 10 terminals. Independently, the probability that any one terminal will require attention during a week is 0⋅10 \cdot 1. Find the probabilities that (iv) 3 or more, terminals will require attention during the next week.
  25. Misc 8 Q.14 (i)
    In a large school, 80% of the pupils like mathematics. A visitor to the school asks each of 4 pupils, chosen at random, whether they like mathematics. (i) Calculate the probabilities of obtaining an answer yes from 0, 1, 2, 3, 4 of the pupils.
  26. Misc 8 Q.14 (ii) (a)
    In a large school, 80% of the pupils like mathematics. A visitor to the school asks each of 4 pupils, chosen at random, whether they like mathematics. (ii) Find the probability that the visitor obtains the answer yes from at least 2 pupils: (a) when the number of pupils questioned remains at 4.
  27. Misc 8 Q.14 (ii) (b)
    In a large school, 80% of the pupils like mathematics. A visitor to the school asks each of 4 pupils, chosen at random, whether they like mathematics. (ii) Find the probability that the visitor obtains the answer yes from at least 2 pupils: (b) when the number of pupils questioned is increased to 8.
  28. Misc 8 Q.15 (i)
    It is observed that, it rains on 12 days out of 30 days. Find the probability that (i) it rains exactly 3 days of week.
  29. Misc 8 Q.15 (ii)
    It is observed that, it rains on 12 days out of 30 days. Find the probability that (ii) it will rain on at least 2 days of given week.
  30. Misc 8 Q.16
    If probability of success in a single trial is 0⋅010 \cdot 01. How many trials are required in order to have probability greater than 0⋅50 \cdot 5 of getting at least one success?
  31. Misc 8 Q.17
    In binomial distribution with five Bernoulli's trials, probability of one and two success are 0⋅40960 \cdot 4096 and 0⋅20480 \cdot 2048 respectively. Find probability of success.