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]
= nCr prq(n-r) , r =
0,1,2,...,n
P[X=r]
= 5Cr (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]
= nCr prq(n-r) , r =
0,1,2,...,n
P[X=r]
= 5Cr ( 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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link to index of other miscellaneous problems on probability of cbse ncert 12th mathematics
index of more problems on baye's theorem for ncert cbse mathematics
index of more problems on baye's theorem for ncert cbse mathematics
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