What is the probability that a deck of cards includes all four suits when randomly selecting five cards without replacement?

To calculate the probability of selecting all four suits when randomly selecting five cards without replacement from a deck of cards, we need to find the number of favorable outcomes and the total number of possible outcomes.

First, let's determine the number of favorable outcomes. To have all four suits represented, we need to select at least one card from each suit.

There are 13 cards in each suit (A, 2-10, J, Q, K), so the number of ways to select one card from each suit is 13^4 (since there are four suits). This accounts for cases where all four suits are chosen in the first four cards.

Next, we need to determine the number of ways to select the fifth card that can be of any suit. Since there are four suits, any of the remaining 13 cards could be chosen. Therefore, there are 13 * 4 = 52 ways to choose the fifth card.

Now, let's calculate the total number of possible outcomes. Initially, there are 52 cards to choose from for the first card, then 51 cards for the second card, 50 cards for the third card, 49 cards for the fourth card, and finally 48 cards for the fifth card. Therefore, the total number of possible outcomes is 52 * 51 * 50 * 49 * 48.

To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:

Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)

Probability = (13^4 * 52) / (52 * 51 * 50 * 49 * 48)

Simplifying the expression, we get:

Probability = (13 * 13 * 13 * 13 * 52) / (52 * 51 * 50 * 49 * 48)

Simplifying further:

Probability = (169 * 169 * 52) / (52 * 51 * 50 * 49 * 48)

Now, we can cancel out some terms:

Probability = (169 * 169 * 2) / (51 * 50 * 49 * 48)

Finally, we can calculate the probability:

Probability ≈ 0.1051 or 10.51%

So, the probability that a deck of cards includes all four suits when randomly selecting five cards without replacement is approximately 0.1051 or 10.51%.

To determine the probability of selecting all four suits when randomly selecting five cards without replacement from a deck of cards, we need to calculate the number of favorable outcomes and the total number of possible outcomes.

Number of favorable outcomes:
To have all four suits represented, we need at least one card from each suit. Let's calculate the number of ways to select one card from each suit, and then consider the remaining cards.

- Number of ways to select one card from each suit:
- There are 13 cards in each suit, so the number of ways to select one card from each suit is 13 * 13 * 13 * 13 = 28561.

- Number of ways to select the remaining card(s):
- After selecting one card from each suit, we have 9 remaining cards to choose from (since we have already chosen one card from each suit).
- We need to select (5 - 4) = 1 more card to complete the selection of five cards.
- Since there are 9 remaining cards and we need to select 1, there are 9 choose 1 = 9 ways to do this.

Therefore, the total number of favorable outcomes is 28561 * 9 = 257049.

Total number of possible outcomes:
The total number of possible outcomes when selecting five cards without replacement from a deck of 52 cards can be calculated using the combination formula:
- 52 choose 5 = (52! / (5! * (52-5)!)) = 2,598,960.

Therefore, the total number of possible outcomes is 2,598,960.

Probability:
To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
- Probability = Favorable outcomes / Total outcomes
- Probability = 257,049 / 2,598,960
- Probability ≈ 0.0989 or 9.89% (rounded to four decimal places).

Therefore, the probability of randomly selecting five cards without replacement and getting all four suits represented is approximately 0.0989 or 9.89%.