Question

Among the 2,598,960 possible five card poker hands from a standard 52 card deck, how many contain at least one spade?

Answer 1

To find the number of five-card poker hands that contain at least one spade, we calculate the complementary number of hands without a spade and subtract that from the total number of possible hands (2,598,960). We determine there are 2,023,203 such hands.

Step-by-step explanation:

To calculate how many five-card poker hands contain at least one spade from a standard 52-card deck, we use the concept of complementary probability. First, we find the number of ways to select a hand with no spades, and then subtract that number from the total number of possible hands.

The total number of five-card hands is a combination of 52 cards taken 5 at a time, which is 2,598,960. There are 39 non-spade cards in the deck, so the number of hands with no spades is a combination of 39 cards taken 5 at a time. Let's calculate this:

1. Number of hands with no spades: C(39, 5)
2. Total possible hands: C(52, 5) = 2,598,960
3. Number of hands with at least one spade: 2,598,960 - C(39, 5)

Using a calculator or formula for combinations, we find that C(39, 5) is 575,757. That is the number of hands without a spade. Therefore, the number of hands with at least one spade is:

2,598,960 - 575,757 = 2,023,203

So, there are 2,023,203 five-card hands that contain at least one spade.

Answer 2

All you need to do is find the probability of getting 0 spades for 5 cards in a row, subtract from 1, and multiply by 2,598,960.

I'll even do the first part: P(0 spades) =
(39/52)*(38/51)*(37/50)*(36/49)*(35/48)