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]
=

_{n}C_{r }p^{r}q^{(n-r) }, r = 0,1,2,...,n
P[X=r]
=

_{5}C_{r }(1/4)^{r}(3/4)^{(5-r) }, r = 0,1,2,...,5
P[all
the five cards are spades ] = P[X=5] =

_{5}C_{5}(1/4)^{5}(3/4)^{(5-5)}=( 1 / 1024 )
P[only
three cards are spades ] = P[X=3]
=

_{5}C_{3}(1/4)^{3}(3/4)^{(5-3)}=(90/1024)=(45/512)
P[none
is a spade ] = P[X=0]
=

_{5}C_{0}(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]
=

_{n}C_{r }p^{r}q^{(n-r) }, r = 0,1,2,...,n
P[X=r]
=

_{5}C_{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]

=

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

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