Question

help please!!!

Use the dartboard below to calculate the expected values in terms of
r1 = 2, bull's eye worth 3 points
r2 = 4, middle ring worth 1 point
r3 = 2sqrt6, outer ring worth -1 point


A.2/3
B.1/6
C.16
D.4/11

Answer 1

we need to see the areas involved, then the probability of landing in each area is the ratio of that area to the total area

Answer 2

sorry , im confused . so what would be my first step to solve this?

Answer 3

i assume the \(r_1=2\) means the inner ring has radius 2, and so the area of the inner ring is
\[4\pi\] but we don't know the probability of landing in that area without knowing the total area of the dart board

Answer 4

do you have a picture that goes along with this question?

Answer 5

no only this :/

Answer 6

ok i am going to assume that this means the whole board has radius \(2\sqrt{6}\) and therefore since the area of a circle is \(\pi r^2\) the area of the dart board is \(\pi (2\sqrt{6})^2=24\pi\)

Answer 7

the inside circle has radius 2, so its area is \(4\pi\) and the ratio of the areas is
\[\frac{4\pi}{24\pi}=\frac{1}{6}\] meaning the probability you land in that area (assuming the dart lands anywhere on the board with equal probability) is \(\frac{1}{6}\)

Answer 8

now we need the probability you land in the 1 point ring
if the radius of that ring is 4, then the area of that ring (of course not counting the center) is
\[16\pi-4\pi=12\pi\] and the ratio of that area to the total is
\[\frac{12\pi}{24\pi}=\frac{1}{2}\]

Answer 9

the probability you end up in the remaining -1 point area should be
\[1-\frac{1}{2}-\frac{1}{6}=\frac{1}{3}\]

Answer 10

to compute the expected value, you multiply the points you get by the probability you get them and add up
so it should be (if i haven't made a mistake along the way, you should check)
\[3\times \frac{1}{6}+1\times\frac{1}{2}-1\times \frac{1}{3}\]

Answer 11

sorry i had to finish dinner , but i cant read what you put at the bottom :/

Answer 12

@satellite73

Answer 13

refresh your browser if you cannot read the math symbols

Answer 14

\[3\times \frac{1}{6}+1\times\frac{1}{2}-1\times \frac{1}{3}\]

Answer 15

i get \(\frac{2}{3}\) which is good because that is one of the choices

Answer 16

thank you! :) it showed up after i refreshed it in big bold text , thanks!

Answer 17

ok hope all steps are clear, this one had quite a few steps

Answer 18

yw