If you are drawing 3 cards onebyone with replacement from a

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.

Solution

Total count of sample space S = 52C1 * 52C1 * 52C1 = (52)^3 = 140608

A (all 3 are red) = 26C1 * 26C1 * 26C1 = 17576 ways ( since there are two clubs of red cards)

B(all 3 are diamonds) = 13C1 * 13C1 * 13C1 = 2197 ways

C(all 3 are king of spades) = 1C1 * 1C1 * 1C1 = 1 way

D(all 3 are of the same color) = 26C1 * 26C1 * 26C1 * 2(either black or red colour) = 35152 ways

E(all 3 are from same suit) = 13C1 * 13C1 * 13C1 * 4 (there are four suits in total)

=> 8788 ways

F(all 3 of them are teh same cards) = 1 * 1 * 1 = 1 way

P(A) = 17576/140608 = 0.125

P(B) = 2197/140608 = 0.015625

P(C) = 1/140608 = 7.1197 * 10^(-6)

P(D) = 35152/140608 = 0.25

P(E) = 8788/140608 = 0.0625

P(F) = 1/140608 = 7.1197 * 10^(-6)

There are 52 - 12 = 40 number cards

G = 40C1 * 40C1 * 40C1 = 64000

P(G) = 64000/140608 = 0.4551