We are tasked with calculating the probability of getting either a "three of a kind," "four of a kind," or a "royal flush" when dealt a 5-card hand from a standard 52-card deck.

Data from the image:

• Royal flush: 4 outcomes
• Four of a kind: 624 outcomes
• Three of a kind: 54,912 outcomes
• Total possible outcomes (5-card hand): 2,598,960

Probability Formula:

The probability of an event occurring is given by:

$P(E) = \frac{\text{Number of favorable outcomes for event E}}{\text{Total possible outcomes}}$

Calculation:

To find the total probability of getting either a "three of a kind," "four of a kind," or a "royal flush," we sum the favorable outcomes of these events:

$\text{Total favorable outcomes} = 4 + 624 + 54,912 = 55,540$

Now, we calculate the probability:

$P = \frac{55,540}{2,598,960} \approx 0.021374$

Final Answer:

The probability of getting a three of a kind, four of a kind, or royal flush is approximately 0.021374 (rounded to six decimal places).

Do you need further details or have any questions?

Here are 5 related questions to explore:

1. What is the probability of getting a straight flush or a full house?
2. How does the probability of getting a "two pairs" compare to a "three of a kind"?
3. What is the probability of getting any hand that is not a "no pair"?
4. How does the total probability of a flush compare to a straight?
5. What is the probability of getting no pair at all in a 5-card hand?

Tip: When calculating probabilities in card games, always consider whether the deck is shuffled after each hand or whether cards are drawn without replacement!