Question

applying the law of large numbers to a carnival game
at a carnival, you spin a spinner that has four equal - sized sections, each a different color: green, yellow, red, and blue. if you land on green, you win 2 points. if you land on yellow, red, or blue, you lose 1 point. write the expected value equation.
$e(x)=\frac{1}{a}(2)+\frac{1}{b}(c)+\frac{1}{d}(e)+\frac{1}{f}(g)$
$a = square$ $x$ $rightarrow4$ $b=square$ $x$ $rightarrow4$
$c=square$ $x$ $rightarrow - 1$ $d=square$ $x$ $rightarrow4$
$e=square$ $x$ $rightarrow - 1$ $f=square$ $x$ $rightarrow4$
$g=square$ $x$ $rightarrow - 1$
the expected value of the number of points won on a spin is $square$

Explanation:

Step1: Determine probabilities and values

The spinner has 4 equal - sized sections. The probability of landing on green is $\frac{1}{4}$, and the value for green is 2 points. The probability of landing on yellow, red, or blue is $\frac{3}{4}$, and the value for each of these is - 1 point. The expected - value formula is $E(X)=\sum_{i}p_ix_i$. Here, $E(X)=\frac{1}{4}(2)+\frac{3}{4}(-1)$.

Step2: Calculate the expected value

First, calculate $\frac{1}{4}(2)=\frac{2}{4}=\frac{1}{2}$, and $\frac{3}{4}(-1)=-\frac{3}{4}$. Then $E(X)=\frac{1}{2}-\frac{3}{4}=\frac{2 - 3}{4}=-\frac{1}{4}$.

Answer:

$-\frac{1}{4}$