
Working out probabilities of shuffling playing cards
By MUNGAI KIHANYA
The Sunday Nation
Nairobi,
08 January 2023
We were playing a
game of cards during the Christmas break and some one asked me if it is
possible to shuffle the deck and get all the cards organised in
ascending order from 2, 3, 4…J, Q, K, to A. and the with suits in the
same order (hearts, diamonds, clovers and spades) for each of the ranks.
Well, the answer is yes, but the probability is miniscule.
Suppose you had just
two cards labelled A and B. The are only two ways of arranging them: AB
and BA. So, whenever you shuffle them you can only get one of these two
arrangements. So, the probability of getting the two cards in
alphabetical order is one out of two possibilities, that is 0.5.
If there are three
cards, A, B and C, then the possible arrangements are ABC, BAC, ACB,
BCA, CAB, and CBA. These are 6 possible outcomes, thus the probability
that, after shuffling you will get, say, ABC, is one out of 6, that is
0.17.
With four cards,
ABCD, the arrangements are: ABCD, BACD, ACBD, BCAD, CABD, CBAD, ABDC,
BADC, ACDB, BCDA, CADB, CBDA, ADBC, BDAC, ADCB, BDCA, CDAB, CDBA, DABC,
DBAC, DACB, DBCA, DCAB and DCBA.
Stop! Don’t count
them yet. Let’s analyse them first. Look at the first six arrangements
from ABCD to CBAD. The first three letters are exactly the same as those
in the possible arrangements of three cards. The only difference is that
now we add card D at the end. In the next series of six patterns the D
card is second from last. Then it is the third from last and finally, it
moves the first position in the last six patterns.
Clearly, then, there
are four possible positions for D: last, second from last, third from
last, and first. Therefore, the number of patterns possible with four
cards is four times those of three cards; that is, 4 x 6 = 24. Now you
can go back and count them!
It is the same for
three cards: the possible arrangements are 3 times the two of the two
cards: 3 x 2 = 6. And four five cards, there will be 5 x 24 = 120
arrangements (24 being the number we have evaluated with four cards).
Can you see the sequence?
Following this
sequence, we find that the number of patterns possible with a full deck
of 52 cards must be 52 x 51 x 50 x 49 x 48 x …x 5 x 4 x 3 x 2. The
answer is a number with 67 zeros! We can confidently say that, in the
500year history of playing cards, nobody has ever shuffled a deck and
got them all in the ascending order.

