five cards are chosen at random from a standard deck of 52 cards. what is the the probability that all five cards are the same suit? express your answer as a common fraction.
To find the probability that all five cards are the same suit, we need to determine the number of favorable outcomes (the number of ways to select five cards of the same suit) and the number of possible outcomes (the total number of ways to select any five cards from the deck).
First, let's determine the number of favorable outcomes. We have four suits in a standard deck: hearts, diamonds, clubs, and spades. We need to choose five cards of the same suit, so we have to choose one of the four suits. Once we choose a suit, there are 13 cards of that suit in the deck, so we can choose five of them in ${13 \choose 5} = \frac{13!}{5!(13-5)!}$ ways.
Now, let's determine the number of possible outcomes. We can choose any five cards from the deck, so there are ${52 \choose 5} = \frac{52!}{5!(52-5)!}$ ways to select five cards from the 52-card deck.
Therefore, the probability that all five cards are the same suit is given by:
\[
P = \frac{\text{{number of favorable outcomes}}}{\text{{number of possible outcomes}}} = \frac{\frac{13!}{5!(13-5)!}}{\frac{52!}{5!(52-5)!}}
\]
Simplifying this expression gives us the probability as a common fraction.
Note: To calculate factorials and combinations, you can use a calculator, software, or online tools.
To find the probability that all five cards are the same suit, we need to determine the total number of favorable outcomes (i.e., all five cards belonging to the same suit) and the total number of possible outcomes (i.e., any five cards from a standard deck).
First, let's calculate the total number of possible outcomes. There are 52 cards in a standard deck, and we need to choose 5 cards at random. So, the total number of possible outcomes is given by the binomial coefficient "(52 choose 5)" or "52C5," which can be calculated as follows:
(52C5) = 52! / (5! * (52-5)!)
= (52 * 51 * 50 * 49 * 48) / (5 * 4 * 3 * 2 * 1)
= 2,598,960
Next, we'll determine the total number of favorable outcomes, which is the number of ways to choose 5 cards from one of the four suits in the deck. Since there are four suits (clubs, diamonds, hearts, and spades), the total number of favorable outcomes is 4 * (13C5), where (13C5) represents the number of ways to choose 5 cards from a single suit.
Using the binomial coefficient again, we can calculate (13C5) as follows:
(13C5) = 13! / (5! * (13-5)!)
= (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1)
= 1,287
Therefore, the total number of favorable outcomes is 4 * 1,287 = 5,148.
Finally, we can find the probability by dividing the total number of favorable outcomes by the total number of possible outcomes:
P(all five cards are the same suit) = 5,148 / 2,598,960
Simplifying the fraction as much as possible, we get:
P(all five cards are the same suit) = 1/507