Question

question 11
the 4 - year old daughter of a friend is doodling on a sketchpad. her parents have placed 9 different colored crayons and 14 different colored marker pens in a small container for her. the little girl is choosing crayons and marker pens from this container randomly to sketch and finish up her drawings. what is the probability that she randomly picks 6 crayons and 4 marker pens to finish up a drawing?
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 items

Total items \(n = 9 + 14=23\). Number of ways to choose \(6 + 4=10\) items from \(23\) is given by combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\). So, \(C(23,10)=\frac{23!}{10!(23 - 10)!}=\frac{23!}{10!×13!}=\frac{23\times22\times21\times20\times19\times18\times17\times16\times15\times14}{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}=1144066\)

Step2: Calculate number of ways to choose crayons and markers

Number of ways to choose \(6\) crayons from \(9\) is \(C(9,6)=\frac{9!}{6!(9 - 6)!}=\frac{9!}{6!×3!}=\frac{9\times8\times7}{3\times2\times1}=84\). Number of ways to choose \(4\) markers from \(14\) is \(C(14,4)=\frac{14!}{4!(14 - 4)!}=\frac{14!}{4!×10!}=\frac{14\times13\times12\times11}{4\times3\times2\times1}=1001\)

Step3: Calculate probability

By multiplication principle, number of favorable cases \(N = C(9,6)\times C(14,4)=84\times1001 = 84084\). Probability \(P=\frac{N}{C(23,10)}=\frac{84084}{1144066}\approx0.0735\)

Answer:

\(0.0735\)