Questions: Now we will calculate the actual probability that all the cards we are dealt in a poker game are of the same suit. We have found that the probability is given as (4 C1 * 13 C5) / (52 C5) Express your answer as a decimal rounded to four decimal places.

Solution Steps

To find the probability that all the cards dealt in a poker game are of the same suit, we need to calculate the given expression: \[ \frac{{ }_{4} C_{1} \times{ }_{13} C_{5}}{{ }_{52} C_{5}} \] where \({ }_{n} C_{k}\) represents the binomial coefficient, which is the number of ways to choose \(k\) items from \(n\) items without regard to order.

1. Calculate \({ }_{4} C_{1}\), which is the number of ways to choose 1 suit from 4 suits.
2. Calculate \({ }_{13} C_{5}\), which is the number of ways to choose 5 cards from 13 cards of the chosen suit.
3. Calculate \({ }_{52} C_{5}\), which is the number of ways to choose 5 cards from 52 cards.
4. Divide the product of the first two results by the third result to get the probability.
5. Round the result to four decimal places.

The number of ways to choose 1 suit from 4 suits is given by the binomial coefficient: \[ { }_{4} C_{1} = 4 \]

The number of ways to choose 5 cards from 13 cards of the chosen suit is: \[ { }_{13} C_{5} = 1287 \]

The number of ways to choose 5 cards from 52 cards is: \[ { }_{52} C_{5} = 2598960 \]

The probability that all the cards dealt are of the same suit is calculated as: \[ P = \frac{{ }_{4} C_{1} \times { }_{13} C_{5}}{{ }_{52} C_{5}} = \frac{4 \times 1287}{2598960} \approx 0.0019807923169267707 \] Rounding this to four significant digits gives: \[ P \approx 0.002 \]

Final Answer