Final answer:
The question is about using the normal distribution to find the probability that a random group of nine casino patrons will average more than $500 in winnings. This involves calculating the z-score and then finding the corresponding probability using standard normal distribution tables or software.
Step-by-step explanation:
The question revolves around the concept of the normal distribution and its use in calculating probabilities of certain events. Specifically, it involves understanding the distribution of winnings and losses among casino patrons in Las Vegas.
The average loss per day is given as $110 with a standard deviation of $700. We are asked to find the probability that a random group of nine people will average more than $500 in winnings in a one-day trip. To solve this, we will use the central limit theorem, which tells us that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough (which it is in our case).
The expected value of the sample mean is the same as the population mean, so it's -$110. However, the standard deviation of the sample mean (standard error) is the population standard deviation divided by the square root of the sample size, which is $700/\(\sqrt{9}\) = $700/3 = $233.33.
We are interested in the probability that the sample mean is greater than $500. To find this, we would calculate the z-score, which is the number of standard deviations away from the mean our value of interest lies, and then refer to standard normal distribution tables or use statistical software to find the corresponding probability.