Respuesta :
Answer:
[tex]\frac{1}{2}[/tex]
Step-by-step explanation:
Given:
A gear bag with 3 bungee cords, 5 socket wrenches, and 2 hammers.
Total number of instruments = 3 + 5 + 2 = 10
Question asked:
What is the probability a person will randomly draw a socket wrench from the bag?
Solution:
As we know:
[tex]Probability =\frac{Favourable \ outcome}{Total\ outcome}[/tex]
First of all we will find favorable outcome for drawing 1 socket wrench from the bag randomly:
Favorable outcome for drawing 1 socket wrench out of cookies 3 bungee cords =[tex]^{3} C_{0}[/tex]
Favorable outcome for drawing 1 socket wrench out of 5 socket wrenches = [tex]^{5} C_{1}[/tex]
Favorable outcome for drawing 1 socket wrench out of 2 hammers = [tex]^{2} C_{0}[/tex]
Thus, total Favorable outcome for drawing 1 socket wrench from the bag randomly: By using:
[tex]^{n} C_{r}=\frac{n!}{(n-r)!\ r!}[/tex]
[tex]^{3} C_{0}[/tex] [tex]\times[/tex][tex]^{5} C_{1}[/tex][tex]\times[/tex] [tex]^{2} C_{0}[/tex]
[tex]\frac{3!}{(3-0)!\ 0!} \times\frac{5!}{(5-1)!\ 1!}\times\frac{2!}{(2-0)!\ 0!}[/tex]
[tex]\frac{3!}{3!\times1} \times\frac{5!}{4!\times1} \times\frac{2!}{2!\times1} \\\\ 1\times\frac{5\times4!}{4!} \times\frac{2!}{2!} \\\\ 1\times5\times1=5[/tex]
Total outcome for drawing 1 socket wrench out of 10 pieces of instruments:
[tex]^{10} C_{1} =\frac{10!}{(10-1)!\times1!} =\frac{10\times9!}{9!} =10[/tex]
Now,
[tex]Probability =\frac{Favourable \ outcome}{Total\ outcome}[/tex]
[tex]=\frac{5}{10} =\frac{1}{2}[/tex]
Thus, the probability that a person will randomly draw a socket wrench from the bag is [tex]\frac{1}{2}[/tex]
