Probablity of no fives left in deck with 26 cards left

Question

In ed thorps book, a man for all markets,he says the probability that no fives are left in a deck when 26 cards are left is 5 per cent.with thirteen cards left it is 30 per cent. Can anyone prove this?

Answer 1

Assuming fair drawing of a shuffled standard deck of cards, all sets of $26$ cards are equally likely to be what remain in the deck.

The probability of no fives being left in the deck is going to be:

$$\dfrac{\binom{4}{0}\binom{48}{26}}{\binom{52}{26}}=\dfrac{46}{833}\approx 0.05522209\dots$$

For thirteen cards left it would be

$$\dfrac{\binom{4}{0}\binom{48}{13}}{\binom{52}{13}}=\dfrac{6327}{20825}\approx 0.303817527\dots$$

Both of these follow trivially from counting principles and/or the hypergeometric distribution.

Answer 2

$$\frac{\binom{4}{4}\binom{48}{22}}{\binom{52}{26}}\approx 5.52\%$$

2.

$$\frac{\binom{4}{4}\binom{48}{35}}{\binom{52}{39}}\approx 30.38\%$$