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Friday, December 30, 2016

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] = nCr prq(n-r) , r = 0,1,2,...,n
P[X=r] = 6Cr ( 1 / 2 )r( 1 / 2 )(6 - r)

P[X=r] = 6Cr ( 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] = nCr prq(n-r) , r = 0,1,2,...,n

P[X=r] = 4Cr (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 12th mathematics and is useful for the students preparing for the board examination 

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