A deck of 52 cards is mixed well and 5 cards are dealt a It

A deck of 52 cards is mixed well, and 5 cards are dealt.

(a) It can be shown that (disregarding the order in which the cards are dealt) there are 2,598,960 possible hands, of which only 1287 are hands consisting entirely of clubs.
What is the probability that a hand will consist entirely of clubs? (Give the answer to six decimal places.)


What is the probability that a hand will consist entirely of a single suit? (Give the answer to six decimal places.)


(b) It can be shown that exactly 63,206 hands contain only spades and hearts, with both suits represented.
What is the probability that a hand consists entirely of spades and hearts with both suits represented? (Give the answer to five decimal places.)


(c) Using the result of Part (b), what is the probability that a hand contains cards from exactly two suits? (Give the answer to five decimal places.)

Solution

a) Probability that hand will consist entirely of clubs = 1287 / 2598960

=0.000495

Probability that hand will consist single suit = (number of cases in which hand will consists entirely club +number of cases in which hand will consists entirely spade + number of cases in which hand will consists entirely diamond +number of cases in which hand will consists entirely heart) / total number of ways to select 5 cards from 52 cards

= (1287 +1287 +1287 +1287) / 2598960

= 0.001981

b)

we are given that total number of ways in which only spade and herats are selected from 5 cards are 63,206

The probability that a hand consists entirely of spades and hearts with both suits =total number of ways in which only spade and herats are selected from 5 cards /  total number of ways to select 5 cards from 52 cards

= 63206/ 2598960

= 0.02432

c)

There are 6 ways of selecting the cards from two suits i.e.

SC, SD, SH, CH, CD, DH   

Where

S= spade

C =club

D =diamond

H= heart.

number of ways of selecting cards from each two suit is  63,206

Therfore probability of selecting cards from the same suit is (63206+63206+63206+63206+63206+63206)/ 2598960

= 0.145918