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)
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(\\(\bar{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(\\(\bar{x}\\) > 0) = 0.3920
(type an integer or decimal rounded to four decimal places as needed.)
(g) compare the results of parts (d) through (f). what lesson does this teach you?
a. the probability of being ahead decreases as the number of games played increases
b. the probability of being ahead increases as the number of games played increases
c. the probability of being ahead decreases as the number of games played decreases
d. the probability of being ahead remains the same as the number of games played varies

Explanation:

We look at the probabilities \(P(\bar{x}>0)\) for different numbers of games. When the number of games is \(n = 180\), \(P(\bar{x}>0)=0.4512\); when \(n = 900\), \(P(\bar{x}>0)=0.3920\). As the number of games \(n\) increases from a smaller value (implied in part (d) if we assume a smaller \(n\) than 180, but even comparing 180 and 900) to a larger value, the probability of being ahead (\(P(\bar{x}>0)\)) decreases.

Answer:

A. The probability of being ahead decreases as the number of games played increases