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).

μ = -0.05
σ = 0.61

(d) what is the probability of being ahead after playing the game 90 times? that is, what is the probability that the sample mean is greater than 0 for n = 90?
p( x > 0 ) = 0.4681
(type an integer or decimal rounded to four decimal places as needed.)

(e) what is the probability of being ahead after playing the game 180 times?
p( x > 0 ) = 0.4512
(type an integer or decimal rounded to four decimal places as needed.)

(f) what is the probability of being ahead after playing the game 900 times?
p( x > 0 ) =
(type an integer or decimal rounded to four decimal places as needed.)

Explanation:

Step1: Calculate the standard deviation for \(n = 900\)

The formula for the standard deviation of the sample mean is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 0.61\) and \(n = 900\), we have \(\sigma_{\bar{x}}=\frac{0.61}{\sqrt{900}}=\frac{0.61}{30}\approx0.0203\).

Step2: Calculate the z - score

The z - score formula is \(z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}\). We want to find \(P(\bar{x}>0)\), so \(\bar{x} = 0\), \(\mu_{\bar{x}}=- 0.05\), and \(\sigma_{\bar{x}}\approx0.0203\). Then \(z=\frac{0 - (-0.05)}{0.0203}=\frac{0.05}{0.0203}\approx2.46\).

Step3: Find the probability

Using the standard normal distribution table, \(P(Z > 2.46)=1 - P(Z\leqslant2.46)\). From the standard - normal table, \(P(Z\leqslant2.46)=0.9931\). So \(P(Z > 2.46)=1 - 0.9931 = 0.0069\).

Answer:

\(0.0069\)