Given a normal deck of # 52 # cards. If I draw five cards from the deck, what is the probability that all five cards are evenly divisible by # 3? # (assume jacks, queens, kings, and aces to not be divisible by # 3#

The card values which are divisible by #3# are #3,6,9# in each suit, So there are #3xx4=12# cards in the deck which meet the requirement.

#P(DDDDD) = P(D)xxP(D)xxP(D)xxP(D)xxP(D)#

However as each card is drawn, it affects the probabilities for the remaining cards. (Assuming the cards are NOT replaced.)

#=12/52xx11/51xx10/50xx9/49xx8/48#

Cancel where possible first:

#=cancel12/cancel52_13xx11/cancel51_17xxcancel10/cancel50_5xxcancel9^3/49xxcancel8^cancel2/cancel48_cancel4^2#

#=33/10829#

The number of five-card draws that have zero cards divisible by 3 in them is #""_39C_5.# This is because, of the 52 cards in the deck, we only count the 39 that are not divisible by 3 as a successful draw. So, of those 39, we wish to choose 5. Hence #""_39C_5.#

Likewise, the number of total five-card draws is #""_52C_5.#

The probability of drawing a five-card hand with no cards divisible by 3 is then the proportion of successful draws out of all total draws.

#"P"("all 5 not divisible by 3")#

#="num. successful draws"/"num. total draws"#

#=(""_39C_5)/(""_52C_5)#

These "C" values are binomial coefficients, so by request, we can leave our answer as this.