Question

in the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2,, 36. to play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. if the number of the slot the ball falls into matches the number you selected, you win $35, otherwise you lose $1. complete parts (a) through (g) below click here to view the standard normal distribution table (page 1) click here to view the standard normal distribution table (page 2) (a) construct a probability distribution for the random variable x, the winnings of each spin. x p(x) 35 0.0263 -1 0.9737 (type integers or decimals rounded to four decimal places as needed.) (b) determine the mean and standard deviation of the random variable x. round your results to the nearest penny μ = σ =

Explanation:

Step1: Calculate the mean ($\mu$)

The formula for the mean of a discrete random variable is $\mu=\sum x\cdot P(x)$.
So, $\mu = 35\times0.0263+(- 1)\times0.9737$.
First, calculate $35\times0.0263 = 0.9205$.
Then, calculate $(-1)\times0.9737=-0.9737$.
Now, $\mu=0.9205 - 0.9737=-0.0532\approx - 0.05$.

Step2: Calculate the variance ($\sigma^{2}$)

The formula for the variance is $\sigma^{2}=\sum(x-\mu)^{2}\cdot P(x)$.
For $x = 35$:
$(35-(-0.05))^{2}\cdot0.0263=(35.05)^{2}\cdot0.0263 = 1228.5025\times0.0263\approx32.3196$.
For $x=-1$:
$(-1 - (-0.05))^{2}\cdot0.9737=(-0.95)^{2}\cdot0.9737 = 0.9025\times0.9737\approx0.8788$.
$\sigma^{2}=32.3196 + 0.8788=33.1984$.

Step3: Calculate the standard deviation ($\sigma$)

Since $\sigma=\sqrt{\sigma^{2}}$, then $\sigma=\sqrt{33.1984}\approx5.76$.

Answer:

$\mu=-0.05$; $\sigma = 5.76$