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)

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}

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.

To calculate the probability of getting a four of a kind in a five-card hand from a standard deck of cards, we need to determine the number of favorable outcomes (getting four cards of the same rank) and the total number of possible outcomes (all possible five-card hands).

1. Favorable outcomes: A four of a kind hand consists of four cards of the same rank and any one card from the remaining 48 cards in the deck. There are 13 ranks in a deck (Ace, 2, 3, ..., 10, Jack, Queen, King), so we have 13 possibilities for the rank. Once we select the rank, there are 4 cards of that rank in the deck, so the number of favorable outcomes is 13 * 4 = 52.

2. Total outcomes: A five-card hand can be formed from the 52 cards in a deck, so the total number of possible outcomes is given by the number of combinations of 52 items taken 5 at a time, which can be calculated using the formula C(n, r) = n! / (r!(n-r)!) where n is the number of items and r is the number of items taken at a time. In this case, we have C(52, 5) = 52! / (5!(52-5)!) = 2,598,960.

3. Probability: The probability of getting a four of a kind hand is the ratio of favorable outcomes to total outcomes. So, the probability is 52 / 2,598,960 ≈ 0.000019998 or approximately 0.002%.

Therefore, the probability of getting a four of a kind in a five-card hand from a standard deck of cards is very low, approximately 0.002%.