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 ofdiamonds.
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 diamonds and spades, with both suits represented.
What is the probability that a hand consists entirely of diamonds and spades 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) Prob that 5 cards are of diamonds = 13C5/52C5 = 1287/2598960

= 0.000495

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b) the probability that a hand consists entirely of diamonds and spades with both suits represented

= 26C5-2(13C5)/52C5

= 63206/2598960

=0.02432

c) the probability that a hand contains cards from exactly two suits

= Prob (spade, dice)+P(dice, hearts)+P(hearts, clubs)+P(spade, hearts)+P(spade, clubs)+P(dice, clubs)

= 4C2(0.02432)

= 6(0.02432)

= 0.145918