Probability times $3$ on clock patience

Question

As a child under $10$ I was at my grandparents house, my papa had taught me how to play 'Clock Patience' and so I'd play it on every visit. On one occasion I managed to play $3$ games in a row to their full conclusion, in other words last card was a King. Upon telling my papa he very angrily told me not to be such a liar. It's likely I may have done something wrong, maybe I didn't shuffle the cards after the first time etc. But it has always had me wondering is there any mathematical formula or the like that could give a probability estimate of the chances that i did in fact manage to complete $3$ in a row?

Answer 1

If you wrote out all the rules for Clock Patience then in principle you could calculate the probability of winning, and of winning three times in a row, and whether not shuffling the deck in between matters. (That's just "in principle". The actual mathematics might be tedious.)

The probabilities will be very small, which is what makes the game interesting.

But your papa was wrong to accuse you. Rare things are just rare, not impossible. If lots of kids all over the world are playing Clock Patience then every once in a long while someone will win three times in a row. That person might be accused of cheating - but didn't.

Edit: I looked up the rules on Wikipedia while @RossMillikan was answering. The computation is easy. Three wins in a row happens about once every $2000$ times. So if on any particular day $10000$ kids play three games then about $5$ of them will win all three. That day you were one of the lucky ones.

Answer 2

The probability that the last card in a shuffled deck is a king is $\frac 1{13}$, so the chance of three in a row is $\frac 1{13^3}=\frac 1{2197}$. That is not too unlikely. This assumes that the deck is well shuffled. It is hard to say if there is a reason to assume a king would be more likely to be the last card if the shuffle is not perfect, as it is not just whether the king is the bottom card of the deck.

Answer 3

According to Wikipedia, the probability of winning Clock Patience is $1\over13$, so, assuming independence from one game to the next, the probability of winning three times in a row is

$$\left(1\over13\right)^3={1\over2197}$$

which is small but will certainly happen on occasion. In particular, that's the probability that you'll win, say, the first three times you play. If you play over and over again, it becomes more and more likely that some three consecutive games will be wins.