Question

question 11
1 pts
there are 12 women and 18 men who have qualified to compete in the world diving championships in florida, usa. not all, but only a select few of this group of women and men can participate in the world championships. a lottery system is used to determine who will compete. what is the probability of randomly selecting 6 women and 6 men to attend the competition?
note: if your answer has up to four decimal places, please enter it in the box below without rounding the number. however, if your answer has more than four decimal places, please round your answer to four decimal places.

Explanation:

Step1: Calculate total number of people

Total women = 12, total men = 18, so total people \( n = 12 + 18 = 30 \). We need to select \( 6 + 6 = 12 \) people in total.

Step2: Calculate combinations for women, men, and total

The combination formula is \( \binom{n}{k}=\frac{n!}{k!(n - k)!} \).

• Combinations of selecting 6 women from 12: \( \binom{12}{6}=\frac{12!}{6!(12 - 6)!}=\frac{12!}{6!6!}=924 \)
• Combinations of selecting 6 men from 18: \( \binom{18}{6}=\frac{18!}{6!(18 - 6)!}=\frac{18!}{6!12!}=18564 \)
• Combinations of selecting 12 people from 30: \( \binom{30}{12}=\frac{30!}{12!(30 - 12)!}=\frac{30!}{12!18!}=86493225 \)

Step3: Calculate probability

Probability \( P=\frac{\binom{12}{6}\times\binom{18}{6}}{\binom{30}{12}} \).
Substitute the values: \( P=\frac{924\times18564}{86493225} \).
Calculate numerator: \( 924\times18564 = 17153136 \).
Then \( P=\frac{17153136}{86493225}\approx0.1983 \).

Answer:

\( 0.1983 \)