some more problems from binomial distribution for cbse ncert class xii mathematics probability

some more problems from binomial distribution for cbse ncert class xii  mathematics probability

4.Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that (i) all the five cards are spades (ii) only 3 cards are spades and (iii) none is a spade?

Let X be the number of spades among the five cards.

Assume X follows Binomial distribution with

n = 5

p = (13/52) = ( 1 / 4 ) [13 spades among the 52 cards ]

q =1 -p

q = (3 / 4)

P[X=r] = _nC_r p^rq^(n-r) , r = 0,1,2,...,n

P[X=r] = ₅C_r (1/4)^r(3/4)^(5-r) , r = 0,1,2,...,5

P[all the five cards are spades ] = P[X=5] =₅C₅(1/4)⁵(3/4)^(5-5) =( 1 / 1024 )

P[only three cards are spades ] = P[X=3] =₅C₃(1/4)³(3/4)^(5-3) =(90/1024)=(45/512)

P[none is a spade ] = P[X=0] =₅C₀(1/4)⁰(3/4)^(5-0) =(243/1024)

9.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 ?

Let X be the number of questions he answered correctly out of the 5 questions. just by guessing

Assume X follows Binomial distribution with

n =5

p = ( 1 / 3 ) [one out the three possible answers is correct and the candidate is guessing ]

q = 1 – p = ( 2 / 3 )

P[X=r] = _nC_r p^rq^(n-r) , r = 0,1,2,...,n

P[X=r] = ₅C_r ( 1 / 3 )^r( 2 / 3 )^{(5 - r)} r = 0, 1 ,..., 5

P[ candidate would get four or more correct answers just by guessing ] = P[X=4] + P[X=5]

= ₅C₄ ( 1 / 3 )⁴( 2 / 3 )^{(5 - 4)} + ₅C₅ ( 1 / 3 )⁵( 2 / 3 )^{(5 - 5)} = ( 11/243 )

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binomial distribution problem for ncert cbse 12th mathematics probability

binomial distribution problem for ncert cbse 12th mathematics probability

1.A die is thrown 6 times. If getting an odd number is a success, what is the probability of (i) 5 successes (ii) at least 5 successes (iii) at most 5 successes?

Let X be the number of successes out of 6 throws

Assume X follows binomial distribution with

n = 6,

p = (3/6) = ( 1 / 2 )

q = 1 -p

q = ( 1 / 2 )

P[X=r] = _nC_r p^rq^(n-r) , r = 0,1,2,...,n

P[X=r] = ₆C_r ( 1 / 2 )^r( 1 / 2 )^{(6 - r)}

P[X=r] = ₆C_r ( 1 / 2 )⁶

P[ 5 successes ] =P[X=5] = ₆C₅ ( 1 / 2 )⁶ = (3 /32)

P[ at least 5 successes ] =P[X=>5]= P[X=5] +P[X=6]= (7/64)

P[ at most 5 successes ] =P[X<=5] =1 - P[X=6]= 1 - ₆C₆ ( 1 / 2 )⁶ = (63 /64)

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2.A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of

two successes.

Let X be the number of doublets out of 4 tosses of a pair of dice.

Assume X follows binomial distribution with

n = 4,

p = (6 / 36) [6 doublets out of 36 possible outcomes in one toss of a pair of dice]

p = ( 1 / 6)

q= 1 -p

q = (5 /6)

P[X=r] = _nC_r p^rq^(n-r) , r = 0,1,2,...,n

P[X=r] = ₄C_r (1/6)^r(5/6)^{(4 - r)} , r = 0,1,2,3,4

P[two successes] = P[X = 2] =₄C₂ (1/6)²(5/6)^{(4 – 2)} = (25/216)

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This problem and the answer is from binomial distribution in the chapter on probability for class xii of cbse ncert 12^th mathematics and is useful for the students preparing for the board examination 

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finding n of binomial distribution given the probability

 finding n of binomial distribution given the probability from miscellaneous 12th cbse ncert mathematics

How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?

Let n be the required number of times the man must toss a fair coin so that the probability of having at least one head is more than 90%

Let X be the number of heads obtained when the coin is tossed n times.

Assuming X follows binomial distribution with

n=n

p = (½)

q = 1 – p = (½)

P[X=r] = _nC_r p^rq^(n-r) , r = 0,1,2,...,n

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positive determinant from miscellaneous probability questions ncert cbse 12th mathematics

positive determinant and electronic assembly with two subsystems question from miscellaneous probability questions ncert cbse 12th mathematics

If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive?

There are 4 entries in the second order determinant ,each of which can be filled in two ways with either 0 or 1

Such determinants can be constructed in 2⁴ ways ( (2^4) ways)

Therefore if S is the sample space n(S) = (2^4) = 16

Let E be the event that the value of the selected determinant is positive. 

Required probability = P(E) = [n(E)] / [n(S)] = 3 /16

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An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known

P(A fails) = 0.2 P(B fails alone) = 0.15 P(A and B fail) = 0.15

Evaluate the following probabilities P(A fails|B has failed) ; P(A fails alone)

P(B fails) =P(A and B fail together ) + P(B fails alone) = 0.15+0.15 = 0.3

P(A fails|B has failed) = [P(A and B fail)] / [P(B fails ) ] = [0.15 / 0.30 =(1/2) = 0.5

P(A fails alone) = P(A fails) - P(A and B fail together ) = 0.2 – 0.15 = 0.05

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hurdle problem from miscellaneous cbse ncert mathematics probability

In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is (5/6). What is the probability that he will knock down fewer than 2 hurdles?

Let X be the number of hurdles the player knocks down out of 10 hurdles

Assume X follows binomial distribution with

n=10

p = 1-(5/6) [because we defined X in terms of the hurdles knocked down ]

p=(1 /6)

q = 1 -p

q =(5/6)

P[X=r] = _nC_r p^rq^(n-r) , r = 0,1,2,...,n

P[X=r] = ₁₀C_r (1/6)^r(5/6)^{(10 - r)} , r = 0,1,2,...,10

P[the player will knock down fewer than two hurdles] = P[ X < 2 ]

P[ X < 2 ] = P[X=0] + P[X=1]

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index of miscellaneous problems on probablity

index of miscellaneous problems on probablity for cbse xii mathematics

problem 2
A couple has two children, Find the probability that both children are males, if it is known that at least one of the children is male.Also find the probability that both children are females, if it is known that the elder child is a female.
solution to miscellaneous problem 2 on probability of cbse class 12 mathematics

problem 3
Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.
solution to miscellaneous problem 3 on probability of cbse class 12 mathematics

problem  4
Suppose that 90% of people are right handed. What is the probability that at most 6 of a random sample of 10 people are right handed?
solution to miscellaneous problem 4 on probability of cbse class 12 mathematics

problem

In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is (5/6). What is the probability that he will knock down fewer than 2 hurdles?

solution to hurdles problem of cbse class 12th probability 

problem

If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive?

solution to positive determinant question from miscellaneous probability problem in cbse ncert xii mathematics

problem

An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known

P(A fails) = 0.2 P(B fails alone) = 0.15 P(A and B fail) = 0.15

Evaluate the following probabilities P(A fails|B has failed) ; P(A fails alone)

 answer to electronic assembly question from miscellaneous probability problem in cbse ncert xii mathematics

problem

How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?

finding the value of n in binomial distribution given the probability question for cbse ncert 12th mathematics

problem
Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of a certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
solution to miscellaneous problem  on probability of cbse class 12 mathematics


problem

Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls.One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black
solution to miscellaneous problem  on probability of cbse class 12 mathematics
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miscellaneous problem 4 on binomial distribution

miscellaneous problem 4 on binomial distribution for cbse xii probability

Suppose that 90% of people are right handed. What is the probability that at most 6 of a random sample of 10 people are right handed?

Let X be the number of people who are right handed out of a sample of 10 people.

Assuming X follows binomial distribution with
n = 10
p = (90/100) = (0.9) [given that 90% of people are right handed]
q = 1 - p = (0.1)

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