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]
= 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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