(25 points) Princesses Lydia and Ricki are attempting to open a casino for the crumb-snatching kids of their neighborhood. Help them out with the following probabilities so they can properly set the wages for their games.
(a) You roll 6 standard six-sided dice. Determine the probability you roll at least one three.
(b) You roll 12 standard six-sided dice. Determine the probability you roll at least two threes.
(25 points) After hacking the analyst department at the Rivers Casino, you and your friends determine that the probability of winning a certain slot jackpot is 3%.
Explain why we can model this game as a binomial experiment. Determine the exact probability that you win the jackpot at least once in 100 tries.
Use the normal approximation to binomial distribution to approximate the probability you win at least 45 times in 5000 tries.
(25 points) A recent study claims that the starting salaries for a college graduate in the actuarial field are normally distributed with a mean of $65,000 and standard deviation of $7,000.
Determine the probability that an individual graduate will have a salary of over $67,000.
Suppose a group of 50 graduates are selected at random. Determine the probability that the mean of this sample will be greater than $67,000.
(25 points) You have just developed a new, energy-efficient battery for mobile phones. Before putting the battery on the market, you want to estimate how long it will last on a single charge. To investigate, you take a random sample of 50 batteries to find a mean runtime of 300 hours. Assuming the population standard deviation is 17 hours, determine the 99% confidence interval of the population mean.