1. What is the probability that you get an even number at the first toss? (Show work and write the answer in simplest fraction form)
2. What is the probability that the sum of the two tosses is at least 7, given that you get an even number in the first toss? (Show work and write the answer in simplest fraction form)
3. If event A is "Getting an even number in the first toss" and event B is "The sum of two tosses is at least 7". Are event A and event B independent? Use mathematical expression to justify your answer.
4. A high school with 1000 students offers two AP courses: Statistics and Calculus. There are 200 students in the Calculus class roster, and 180 students in the Statistics class roster. We also know that 100 students enroll in both courses.
(a) Find the probability that a randomly selected student takes neither AP course. (Show work and write the answer in simplest fraction form)
(b) Let event A be that a randomly selected student takes AP Statistics course, and event B be that a randomly selected student takes AP Calculus course. Are A and B independent? Use mathematical expression to justify your answer.
5. The steering committee of UMUC Green Solutions Team consists of 3 people. 10 members are interested in serving in the committee. How many different ways can the committee be selected? (Show work)
6. A combination lock uses three distinctive numbers between 0 and 49 inclusive. How many different ways can a sequence of three numbers be selected? (Show work)
7. Imagine you are in a game show. There are 10 prizes hidden on a game board with 100 spaces. One prize is worth $50, two are worth $20, and another seven are worth $10. You have to pay $5 to the host if your choice is not correct. Let the random variable x be the winning.
(a) Complete the following probability distribution. (Show the probability in fraction format and explain your work)
(b) What is your expected winning or loss in this game? Be specific in your answer whether it's winning or loss. (Show work and round the answer to two decimal places)
(c) What is the standard deviation of the probability distribution? (Show work and round the answer to two decimal places)
8. Mimi plans to make a random guess at 10 true-or-false questions. Answer the following questions:
(a) Assume random number X is the number of correct answers Mimi gets. As we know, X follows a binomial distribution. What is n (the number of trials) , p (probability of success in each trial) and q (probability of failure in each trial)?
(b) What is the probability that she gets a passing grade; in other words, she gets at least 6 correct answers? (Show work and round the answer to 4 decimal places)
(c) How many correct answers can she expect to get? (Hint : What is the expected value of the binomial distribution?) (Show work and round the answer to 2 decimal places)
9. Men's weights are normally distributed with mean 170 pounds and standard deviation 20 pounds.
(a) Find the probability that a randomly selected man has a weight between 150 and 200. (Show work and round the answer to 4 decimal places)
(b) What is the 75th percentile for men's weight? (Show work and round the answer to 2 decimal places)
(c) If 64 men are randomly selected, find the probability that sample mean weight is greater than 175. (Show work and round the answer to 4 decimal places)
(For Questions 10, 11 & 12) Justify your answers for full credit.
10. A and B are two disjoint events. If P(A) = 0.5, and P(B) = 0.4, then P(A AND B) is
(a) 0.9
(b) 0.2
(c) 0
(d) cannot be determined
11. A random variable X follows a continuous distribution, which is symmetric to 0. If P(X ≤-2)= 0.2, then which of the following statements are true?
(a) P(X≥2)=0.8
(b) P(X≥2)=0.2
(c) P(X≥-3)≤0.8
(d) P(X≥2)+ P(X<2)=1
12. A is an event, and Ac is the complement of A. Which of the following statements are true?
(a) P(A AND A^c )=1
(b) P(A AND A^c )=0
(c) P(A OR A^c )=1
(d) P(A OR A^c )=0