Question

What is the probably that a five-card hand dealt from a standard deck of cards will include four cards of the same alue? (This is called a "four of a kind" in poker)

Answer 1

figure out how many cards are in a deck {52} then take the number of cards in a deck and compare it to the numbers of cards in a four of a kind {4}

Answer 2

The number of possible ways to deal 5 cards from a standard deck is 52-choose-5 = 52!/5!(52-5)! where ! means factorial. This will be the denominator in your probablity calcuator.
Now for the numerator. There are 13 possible choices for your 4-of-a-kind (A,2,3,..Q,K). That takes care of the 4-of-a-kind part. Your hand still has one more card. There are 48 other cards. So, the number of ways to have 4-of-a-kind in a 5-card hand is 13*48. This is your numerator.