Question

I am dealt a hand of 13 playing cards from a deck of 52. How many bits of information do I need to encode the contents of my hand?

Answer 1

The probability of being dealt a 13-card hand without any hearts is calculated using combinations. It is the ratio of the number of ways to choose 13 cards from the non-heart cards to the number of ways to choose any 13 cards from the entire deck.

Step-by-step explanation:

Calculating the Probability of a No-Heart Hand

To calculate the probability of being dealt a 13-card hand from a standard 52-card deck without a single heart, we can use the concept of combinations. Initially, we have 39 cards that aren't hearts (since there are 13 cards in each suit and we're excluding hearts). The number of ways to choose 13 cards from these 39 is calculated as a combination:

C(39, 13) = 39! / [13!(39 - 13)!]

Also, the total number of ways to choose any 13 cards from a 52-card deck is:

C(52, 13) = 52! / [13!(52 - 13)!]

So, the probability of drawing a hand without any hearts is:

P(no hearts) = C(39, 13) / C(52, 13)

This provides the likelihood of selecting all 13 cards from the 39 non-heart cards.

Note: '!' indicates factorial, meaning the product of all positive integers up to that number. For example, 5! = 5 x 4 x 3 x 2 x 1.