Question

question 13
4 pts
a semi - truck for a recreational cycling group carries 12 different brand mountain bikes and 17 different brand ten - speed bikes. a group of member cyclists go on a weekend recreational cycling trip, and each receives a random pick of the bicycles from the truck. what is the probability that 5 mountain bikes and 5 ten - speed bikes were distributed to the cyclists?
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 the total number of bikes

The total number of bikes is \(12 + 17=29\) bikes. The number of ways to choose \(5 + 5=10\) bikes out of \(29\) is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 29\) and \(r = 10\). So, \(C(29,10)=\frac{29!}{10!(29 - 10)!}=\frac{29!}{10!19!}\)

$$
LATEXBLOCK0
$$

Step2: Calculate the number of ways to choose 5 mountain bikes and 5 ten - speed bikes

The number of ways to choose \(5\) mountain bikes out of \(12\) is \(C(12,5)=\frac{12!}{5!(12 - 5)!}=\frac{12\times11\times10\times9\times8}{5\times4\times3\times2\times1}=792\)
The number of ways to choose \(5\) ten - speed bikes out of \(17\) is \(C(17,5)=\frac{17!}{5!(17 - 5)!}=\frac{17\times16\times15\times14\times13}{5\times4\times3\times2\times1}=6188\)
By the multiplication principle, the number of ways to choose \(5\) mountain bikes and \(5\) ten - speed bikes is \(C(12,5)\times C(17,5)=792\times6188 = 4900904\)

Step3: Calculate the probability

The probability \(P\) is the number of favorable outcomes divided by the number of total outcomes. So \(P=\frac{C(12,5)\times C(17,5)}{C(29,10)}=\frac{4900904}{10015005}\approx0.4894\)

Answer:

\(0.4894\)