Question

We draw cards, one by one, without replacement, from a deck of 52 cards. Calculate the probability that the first ace will appear in the k-th draw, if we know that the n-th card was a spade, and the m-th card was not a club. k=42,m=18,n=4

Answer 1

so according to the given condition

k th draw is first ace appears

m th draw is spade

n th draw is not a club

given k=42 ,m=18, n=4.

since it is given that first ace appears 42 th draw

so ace is rejected for the 41 draws

its probability is 48/52=12/13  

since there would be 4 aces which are rejected

the conditions for all draws except 4th and 18th draw

for 4th draw it is 13 spades in the spades 1 is ace

so total 12/52 for 4th draw

now for m=18 th draw is not a club the remaining are 39 non spades in that 3 are ace hence totally 36 are not spades and not aces

probability for 18 th draw is 36/52

hence the probability for 42nd draw is 4/52

the total probabilty is (48/52)^39 *(12/52)*(36/52)*(4/52) =  0.0005417658