Respuesta :
The bag contains:
2 gold marbles
8 silver marbles
29 black marbles
Prizes:
$4 if it's gold
$2 if it's silver
$1 if it's black
We can calculate the expected value using the following formula:
[tex]E(x)=\sum ^3_{i\mathop=1}x_i\cdot P(x_i)[/tex]Where xi is the money you can get for each marble and P is its probability
Gold marble:
x = $4
P(x) =
[tex]P=\frac{2}{2+8+29}=\frac{2}{39}[/tex]then, x*p(x) is
[tex]4\cdot\frac{2}{39}=\frac{8}{39}[/tex]Silver marble:
x = $2
P(x) =
[tex]P=\frac{8}{2+8+29}=\frac{8}{39}[/tex]then, x*p(x) is
[tex]2\cdot\frac{8}{39}=\frac{16}{39}[/tex]Black marble:
x = $-1
P(x) =
[tex]P=\frac{29}{2+8+29}=\frac{29}{39}[/tex]then x*p(x) is
[tex]-1\cdot\frac{29}{39}=-\frac{29}{39}[/tex]Finally, we need to add all those x*p(x) values we got before to find the expected value, this is:
[tex]E(x)=\sum ^3_{i\mathop{=}1}x_i\cdot P(x_i)=\frac{8}{39}+\frac{16}{39}-\frac{29}{39}=-\frac{5}{39}\cong-0.1282[/tex]Answer: your expected value if you play this game is -0.1282
