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 500-year history of playing cards, nobody has ever shuffled a deck and got them all in the ascending order.