Question

question 10 4 pts a stack of 9 red cards and 10 green cards is shuffled and placed on a table. the cards are distinguished from each other by a number that has been written on each one. assuming that the order of selecting the cards does not matter, what is the probability of randomly selecting 3 red and 5 green cards from this stack. note: if your answer has up to four decimal places, please enter it below without rounding the number. but if your answer has more than four decimal places, round the number to four decimal places.

Explanation:

Step1: Calculate total number of cards

Total number of cards \(n=9 + 10=19\)

Step2: Calculate combination of selecting 3 red cards from 9 red cards

Using combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\), for \(n = 9,k=3\), we have \(C(9,3)=\frac{9!}{3!(9 - 3)!}=\frac{9\times8\times7}{3\times2\times1}=84\)

Step3: Calculate combination of selecting 5 green cards from 10 green cards

For \(n = 10,k = 5\), \(C(10,5)=\frac{10!}{5!(10 - 5)!}=\frac{10\times9\times8\times7\times6}{5\times4\times3\times2\times1}=252\)

Step4: Calculate combination of selecting 8 cards from 19 cards

For \(n=19,k = 8\), \(C(19,8)=\frac{19!}{8!(19 - 8)!}=\frac{19\times18\times17\times16\times15\times14\times13\times12}{8\times7\times6\times5\times4\times3\times2\times1}=75582\)

Step5: Calculate the probability

Probability \(P=\frac{C(9,3)\times C(10,5)}{C(19,8)}=\frac{84\times252}{75582}\approx0.2807\)

Answer:

\(0.2807\)