# Probability of matching a hand of 13 random cards with cards dealt from a shuffled deck up to position 35?

You are given a random assortment of 13 cards from one deck. Another deck is shuffled and cards dealt and called out from the top, the aim is to match all 13 of your cards before reaching 35 cards down into the deck. What is the probability of achieving this?

I realise this is closely related to something like the Birthday Paradox, but I can't seem to use to probability calculator on this site to work out the solution: https://betterexplained.com/articles/understanding-the-birthday-paradox/

The reason I'm asking is because this is a regular thing that happens like a bingo in a pub quiz I frequent and nobody ever wins the prize, which seems unlikely in so many months of playing with a pub full of people all with their own individual random assortment of 13 different cards. After the position of 35 is reached the rest of the deck is read out until someone obviously eventually matches up their cards first, but only win what's in the money pot for the night, rather than big money prize. Me and my friends have worked out various ways to fix the game, but it may be the probability is entirely not in anyone's favour, and the number 35 has been chosen very precisely odds-wise.

Hope someone can provide a solution.

• $C(39,22)/C(52,35)$ = (number of good combinations of 35 cards) / (number of combinations) – Ned Jan 4 at 14:26

If I understand well then your question can be translated into: "if $$13$$ cards are labeled and then after shuffling $$35$$ cards are drawn what is the probability that all labeled cards are among them?"
The answer is: $$\frac{\binom{35}{13}}{\binom{52}{13}}=\frac{35!39!}{22!52!}=\frac{23}{40}\times\cdots\times\frac{39}{52}\approx0.002325$$So a very small chance.