Question

Among seven cards numbered 1, 2, ... , 7, two are drawn with replacement. What is the probability that
a. Both cards are different?
b. Second card has a number larger than first card?

Answer 1

a) A - first card, B - second card

P(A != B) = P(A=1| B!=1)*P(B!=1) + ... + P(A=7| B!=7)*P(B!=7) = 1/7 * 6/7 * 7 = 6/7

b) A = 1 => B = (2,3,4,5,6,7) => P(B>A | A = 1) = 6/7

A = 2 => B = (3,4,5,6,7) => P(B>A | A = 2) = 5/7

A = 3 => B = (4,5,6,7) => P(B>A | A = 3) = 4/7

A = 4 => B = (5,6,7) => P(B>A | A = 4) = 3/7

A = 5 => B = (6,7) => P(B>A | A = 5) = 2/7

A = 6 => B = (7) => P(B>A | A = 6) = 1/7

A = 7 => P(B>A | A = 7) = 0

TOTAL P = P(B>A | A = 1)*P(A = 1) + ... + P(B>A | A = 7)*P(A = 7) = (6+5+4+3+2+1)/49 = 21/49