Question

A 5-card hand is dealt from a perfectly shuffled deck of playing cards. What is the probability of each of the following events:

a) The hand has at least one club

b) The hand has at least two cards with the same rank

c) The hand has exactly one club or exactly one spade

d) The hand has at least one club or at least one spade

Answer 1

Answer:-

Given That:-

A 5-card hand is dealt from a perfectly shuffled deck of playing cards. What is the probability of each of the following events:

number of way to choose 5 cards

from 52 cards = .

(a) first we find prob.of no club.

So remaining hearts ,spades and diamonds =36

So number of ways to choose 5 cards =

So probability of atleast one club

(rounded to 4 decimal places )

(b) No card of same rank .

So we choose 5 ranks among 13 ranks

Now in each rank there are 4 cards and we choose 1 So number of ways

So probability of at least two cards of same rank

(c) choose one club from 13 clubs in 13 ways   

choose one spade  from 13 clubs in 13 ways

and Now choose one remaining 3 cards   from 26 remaining cards

So prob

(d) (at least 1 club or at least one spade) c=no club & no spades

So .choose 5 cards from 26 in

So required probability =