Yves and some friends are playing a fair game in which 18 cards are dealt to 6 players. One of the cards is a queen. The player who receives the queen goes first. What is the probability that Yves will go first?
To find the probability that Yves will go first, we need to determine the total number of possible outcomes and the number of favorable outcomes.
The total number of possible outcomes can be found by considering the number of ways the 18 cards can be distributed among the 6 players. This can be calculated using the concept of combinations, specifically 18 choose 6, denoted as C(18, 6). Using the formula for combinations:
C(18, 6) = 18! / (6! * (18 - 6)!)
Simplifying this expression gives us:
C(18, 6) = 18! / (6! * 12!)
The favorable outcome for Yves going first is that the queen is dealt to him. Since there is only one queen, the number of favorable outcomes is 1.
Now, we can calculate the probability:
Probability = (Number of favorable outcomes) / (Number of possible outcomes)
Probability = 1 / C(18, 6)
Using a calculator or math software, calculate the value of C(18, 6) as:
C(18, 6) = 18! / (6! * 12!) = 18564
So, the probability that Yves will go first is:
Probability = 1 / 18564 ≈ 0.00005385596
Therefore, the probability is approximately 0.0000539 or 0.00539%.