Question

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`

Answer 1

`=33/10829`

Explanation:

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`

Answer 2

Answer to the second part: `(""_39C_5)/(""_52C_5)`

Explanation:

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.