91Ó°ÊÓ

Let's call 'X' the random variable that represents the winning amount on a $1 bet. The following is the probability distribution of X:

The following table demonstrates that,

$$\begin{gathered} \underset{}{∑}P(X=x)=0.474+0.526 \\ =1 \end{gathered}$$

The variable 'X' is a finite population variable, as is the case for discrete random variables.

$\underset{}{∑}P(X=x)=1$

As a result, the probability distribution given is correct.

The random variable X's expected value will be

$$\begin{gathered} E\left(X\right)=\underset{}{∑}xP(X=x) \\ E\left(X\right)=\left[\right(1×0.474)+(-1×0.526\left)\right] \\ E\left(X\right)=(0.474-0.526) \\ E\left(X\right)=-0.052 \end{gathered}$$

The random variable X is expected to have a value of -0.052. As a result, on average, 5.2 cents will be lost per play.

If $1 is bet 100 times on red,

$$\begin{gathered} 5.2cent×100 \\ \$5.2 \end{gathered}$$

If $1 is bet 1000 times on red,

$$\begin{gathered} 5.2cent×1000 \\ \$52 \end{gathered}$$

Roulette is not a profitable game because the chances of winning are extremely low.