Probability and Statistics

Coming home from work, you always seem to hit every single light. Throughout time, you calculate the odds of making it through any light to be 0.2. On any given night, how many lights can you expect to hit before finally making it through one, and with what standard deviation? Finally, what's the probability of the third light you come across being the first one that is green?

A game is played where a player rolls a fair six-sided die until a 1 is rolled. What is the expected mean and standard deviation of the number of rolls?

Instead of a six-sided die, assume a 12-sided die is used in the game. Calculate the mean and standard deviation of the number of rolls until a 1 is rolled.

How many students received at most a score of 69 on the exam?

How many students received a score of at least 80 on the exam?

How many students received a score between 60 and 90 (inclusive)?

Make a frequency table showing the shoe sizes of students in the class using intervals that accommodate decimal values, such as 4 to 6, 6 to 8, 8 to 10, and 10 to 12.

Using the frequency table created, make a histogram of the shoe sizes.

Use the given histogram of winning speeds at the Daytona 500 to answer the following questions:

a) Which interval contains the most data values?

b) How many winning speeds are less than 140 miles per hour?

c) How many winning speeds are at least 160 miles per hour?

Conduct a hypothesis test to determine if a low-fat diet leads to greater weight loss compared to a control group, with a significance level of 5%.

Helen wishes to know whether giving away free stickers will increase her chocolate sales. She tests this by offering free stickers on some days and not on others, then compares the mean sales of the two groups using a t-test for the difference of two means.

Find the probability that all observations in a random sample from an exponential distribution with mean 3 have a value greater than 4.

Calculate the mean and variance of a linear combination of independent random variables given their means and variances.

For independent random variables with given means and variances, find the mean and variance of the linear combination $aX + bY$, and analyze the effect of a given covariance.

Given a product space where $\Omega = \Omega_1 \times \Omega_2$, show that the random variables $X$ and $Y$ are independent when $X$ is a function of $\omega_1$ and $Y$ is a function of $\omega_2$.

Find the probability of rolling a prime number and flipping heads.

Find the probability of choosing both Jokers without replacing the first Joker before the second selection.

Find the probability of choosing a Heart, putting it back, and then choosing a Spade.

Given two events A and B, determine if they are mutually exclusive or independent. Explain your reasoning.

You took a sample of students and found a 95% confidence interval for the true mean amount of sleep to be 7.5 to 8.5 hours. How would you interpret this?