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 diamonds.
What is the probability that a hand will consist entirely of diamonds? (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 hearts and diamonds, with both suits represented.
What is the probability that a hand consists entirely of hearts and diamonds 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)

P(a hand will consist entirely of diamonds)

=(No. of ways a hand consisting entirely of diamonds can be selected)/(No.of possible hands)

=1287/2598960

=0.000495

P(a hand will consist entirely of a single suit)

= P(a hand will consist entirely of diamonds)+P(a hand will consist entirely of hearts)+P(a hand will consist entirely of clubs)+P(a hand will consist entirely of spades)

= 0.000495+0.000495+0.000495+0.000495 (Since all the four probabilities are same)

= 0.001980

(b)

P(A hand contains only hearts and diamonds, with both suits represented)

=63206/2598960

=0.02432

(c)

P(a hand contains cards from exactly two suits)

=6*0.02432           (Since any suits can be selected out of 4 suits in 6 ways)

=0.14592