Drawing two cards from a modified deck without replacement, there are 42 cards in this modified deck of 42 cards and therefore, = nCr (42,2) = The total number of ways to draw two cards is 861 There are 11 ranks in this deck of cards since 9s and 10s have been removed and as a result there are two remaining aces. Each rank consists of 4 cards each and two must be chosen. Therefore, you calculate the number of ways of selecting the two aces and getting a pair of the 10 remaining ranks . There are two aces in this deck and you must draw 2, = nCr ( 2,2) = 1 There are 4 cards for each rank and you must draw 2, multiply by the remaining number of ranks (10) to get the amount. = nCr (4,2) X 10 = 60 Sum the two results together to get the total number of pairs = 60+1= The total number of pairs in this deck is 61. To get the final result, divide the total number of ways drawing two cards by the total number of pairs in this modified deck of cards. = 61861 = 0.0708478513357 - probability of drawing a pair in the modified deck without replacement. P(A) = 80061 = 13.114754098360656 - Probability of not drawing a pair in the modified deck without replacement P(A') Odds against : P(A') : P(A) = The odds against drawing a pair in this modified deck are 800:61 or 13.1147541