Question

in the game of roulette, a wheel consists of the ball falls into matches the number you selected, you win $35 38 slots numbered 0, 00,1,2, 3 6. To play the game, a metal ball is spun around the wheel and is allowed to fall i otherwise you lose $1. Compiete parts (a) through (a) below nto one of the numbered slots. if the number of the slot (a) Construct a probability distribution for the random variable X, the winnings of each spin 35 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 (e) Suppose that you play the game 100 times so that n 100. Describe the sampling distribution of x, the mean amount won per game The sample mean x is What are the mean and standard deviation of the sampling distribution of x? Round your results to the nearest penny is greater than O for n 100 (d) What is the probability of being ahead aher playing the game 100 smes? That is, what is the probabilty hat the sample mean Type an integer or decimal rounded to four decimal places as needed)

Answer 1

a) $p(X=35)={1\over 38}\approx 0.0263$    as there are 38 slots and we choose one

$p(X=-1)={37\over 38}\approx 0.9737$

The table is

X	P(X)
35	0.0263
-1	0.9737

b) Mean and standard deviation is given as

μ E(X)--1 × 0.9737 + 35 × 0.0263ー-0.0532 ~-0.05

Also

$E(X^2)=(-1)^2\times 0.9737+(35)^2\times 0.0263=33.1912\approx 33.19$

Hence

$\sigma=\sqrt{E(X^2)-[E(X)]^2}=\sqrt{33.19-(-0.05)^2}\approx 5.76$

c) The sample mean $\bar{x}$ is approximately normal

$\mu_{\bar{x}}=\mu=-0.05$

and $\sigma_{\bar{x}}={\sigma\over \sqrt{100}}\approx 0.58$

d) Using normal table we get

$\left P(\bar{x}>0)=P(Z>{0-0.05\over 0.58})=P(Z>-0.09)=0.5359$

Where Z is standard random variable

e) for 200 times $\sigma_{\bar{x}}={\sigma\over \sqrt{200}}\approx 0.41$

$\left P(\bar{x}>0)=P(Z>{0-0.05\over 0.41})=P(Z>-0.12)=0.5478$

f) For 1000 ties $\left \sigma_{\bar{x}}={\sigma\over \sqrt{1000}}\approx 0.18$

$\left P(\bar{x}>0)=P(Z>{0-0.05\over 0.18})=P(Z>-0.28)=0.6103$

g) Option B is correct