# Odds for high in the card game pitch

In the card game Pitch how do I calculate when my opponents have 12 cards out of 52 if they have the Ace, King or Queen of a suit? I assume there is only about a 22% chance of the Ace being in their hand, but I don't know how to add the other two cards.

I want my Jack to be the high card, or my partner to have it, we pay double for moons and I'm wanting to know if I should moon with just the jack for high, with 3 cards out there against me.

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## 1 Answer

You have 6 cards (out of 52) and you want to know if another set of 12 (from the same 52) have at least one of three particular cards which you do not have. It is easier to work out the probability they do not have any of the three, which is

$$\frac{43}{46}\times\frac{42}{45}\times\frac{41}{44}\times\frac{40}{43}\times\frac{39}{42}\times\frac{38}{41}\times\frac{37}{40}\times\frac{36}{39}\times\frac{35}{38}\times\frac{34}{37}\times\frac{33}{36}\times\frac{32}{35}$$

$$= \frac{34 \times 33 \times 32}{46 \times 45 \times 44} \approx 0.39$$

so subtracting this from 1 (and multiplying by 100 to get a percentage) means your opponents have a chance of about 61% of having at least one of the three particular cards.

Even if they do have one or two of these cards, it is possible, though less likely, that your partner has an even higher one.

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