# Combination versus fraction

This happens to be about poker but I would like to ask it in a general way

There are 47 cards remaining in the deck with 2 cards to come
Of the 47 cards 9 make my hand - I only need 1 but 2 two also makes my hand

$$\frac { \binom{9}{1} \binom{38}{1} + \binom{9}{2} } { \binom{47}{2} } = 0.349676226$$

Pretty sure that is the correct number as it is posted in poker odds

Tried to get that same number with fractions

The first card is 9/47 = 0.191489362

Second card
Two possibilities - first card hit or not

• first card hit: (9/47)*(8/46) = 0.03330249

• first card did not hit: (1 - 9/47)*9/46 = 0.158186864

Add those up 0.191489362 + 0.033302498 + 0.158186864 = 0.382978723

That is not the same number as combinations 0.382978723 != 0.349676226
My question why are those not the same result?

With fractions this matches 9/47 + (1 - 9/47)*9/46 = 0.349676226
But that does not make sense to me as it seems to me that should be the chance of 1 card hitting excluding the chance of 2 cards hitting

If I go at it 1 - not then it matches
1 - ((47-9)/47*(46-9)/46)
= 1 - (1 - 9/47)*(1 - 9/46)
= 1 - 1 + 9/47 + 9/46 - 9*9/47/46
= 9/47 + 9/46(1 - 9/47)

From the answer I found my mistake - this matches combinations

• first card hits and second does not (9/47)(46-8)/46 = 0.158186864
• first and second (9/47)(8/46) = 0.033302498
• first card does not hit and second does ((47-9)/47)(9/46) = 0.158186864