2 If you are drawing 3 cards onebyone with replacement from

(2) If you are drawing 3 cards (one-by-one with replacement) from a shuffled deck of 52

cards, what is the total count of the sample space S? If A ={all 3 are red}, B= {all 3 are

diamonds}, C = {all 3 are the king of spades}, D = {all 3 are of the same color}, E = {all

3 are from the same suit}, F = {all 3 of them are the same card}, find P(A), P(B), P(C),

P(D), P(E), and P(F). If G = {all 3 are number cards}, check which of the events

mentioned above is (are) independent of G. [Note: In order to check for independence,

you don

Solution

While drawing cards one by one with replacement the first card can be drawn in 52 ways and since it is replaced mean after selection it is returned to the pack. That makes again all 52 available for selection of 2nd card. Same goes for the 3rd. So in totality we can select 3 card in 52*52*52=140608 ways this is the total count of sample space.

D={all 3 are of the same colour}={all are black}+{all are red}. Now there are 26 red and 26 are black. If all 3 are of same colour then they can be chosen in 2*26*26*26=2*17576. P(A)=2*17576/140608=0.250

B={all 3 are diamonds}. There are 13 diamonds in a pack. So if all 3 are diamonds they can be chosen in 13*13*13=2197 ways. P(B)=2197/140608=.0156

About C , I think you wrongly wrote \"all three are kings of spades\" instead of \"all three are kings\". There is only 1 king of spade. So the event you wrote turns out to be a null event hence it will have probability 0. I assume you want the event \"all 3 are kings\". There are 4 kings 3 kings can be chosen in 4*4*4=64 ways. P(C)=64/140608=.00046.

A= {all are red}=26 red cards are there. 3 can b chosen in 26*26*26*=17576 ways. P(A)=17576/140608=.125

E={all are frm same suit}= There are 4 suits each has 13 cards. First 1 out of 4 suits can be chosen in 4 ways. In the chosen suit there are 13 cards out of them 3 can be chosen in 13*13*13=2197 ways. P(E)=4*2197/140608=.0625.

Regarding F={all of them are same cards} I don\'t understand the meaning. All 3 are same? by what? by number or by colour?

G={all 3 are number cards}. C={all three are kings} is independent of G because king cards are nt number cards.