You are at a table at a casino with a gambling game. In the game you spin two spinners. Spinner #1 has 4 equal spaces numbered 1,2,3,4. Spinner #2, has only two equal spaces with the numbers 1 and 3.To play the game, you have to pay $4 dollars. Regardless of whether you win or lose you have to pay the $4. There are two different bets you can make that are related to the sum of the two spins (number from Spinner #1 + number from Spinner #2).Shown below are the two different bets and amounts associated with winning each.

Bet 1: You win $8 if the sum of Spinner #1 and Spinner #2 is less than 5

Bet 2: You win $12 if the sum of Spinner #1 and Spinner #2 is greater than 5

What is the sample space? Hint: this is the number of different outcomes where order does matter. So getting a 1 on Spinner #1 and 3 on Spinner #2 is a distinct outcome from the reverse.

What is the probability of winning Bet 1? What is the probability of winning Bet 2?

What is the expected value of Bet 1? What is the expected value of Bet 2? Which is the better bet? Would you break even (i.e. not lose any money) with either bet?

For Bet 2, how much would you have to earn to exactly break even (neither lose nor win money in expectation)?

Now imagine for Bet 1 you only have to pay the $4 to play if you lose. What is the expected value of Bet 1 now? Would you break even?

6. Let X1, X2, X3 be a random sample from the uniform distribution on the

interval (0, 1). What is the probability that the sample median is less than

0.4?

7. Let X1, X2, X3, X4, X5 be a random sample from the uniform distribution

on the interval (0, ?), where ? is unknown, and let Xmax denote the largest

observation. For what value of the constant k, the expected value of the

random variable kXmax is equal to ??

8. A random sample of size 16 is to be taken from a normal population having

mean 100 and variance 4. What is the 90th percentile of the distribution of

the sample mean?

Question 2 The probability that a freshman passes the course Business Statistics in the first academic year is estimated to be 0.74. The probability that a freshman passes the course Probability in the first academic year is estimated to be 0.25. The probability that a freshman passes the course Business Statistics or the course Probability in the first academic year is estimated to be 0.89. Are passing Business Statistics and passing Probability in the first academic year mutually exclusive? Explain statistically. b. Are passing Business Statistics and passing Probability in the first academic year independent? Explain statistically. What is the probability that a freshman will pass Business Statistic but not Probability in the first academic year? d. What is the probability that a freshman will pass neither Business Statistics nor Probability in the first academic year?Mailings Review View Help Search av AaBbCcDc AaBbCcDo AaBbCc AaBbC v 1 Normal 1 No Spac... Heading 1 Heading Paragraph Styles Instructions: Show your work in calculating each z-value rounded to 2 decimal places o On the blank normal curve, label the horizontal axis with the appropriate z-values o Shade the area under the normal curve that corresponds to the probability you are trying to determine o Show any calculations necessary to determine the final answer o Final answers can be given as decimal numbers to 4 decimal places or percent numbers to 2 decimal places, example: 0.2448 or 24.48% Circle your final answers The average score on a statistics exam was 78.0 with a standard deviation of 5.0. The grades follow the normal distribution. a. What statistics a student is selected at random, what is the probability they achieved a grade greater than 65.0? b. What statistics a student is selected at random, what is the probability they achieved a grade between 78.0 and 85.0? C. What statistics a student is selected at random, what is the probability they achieved a grade less than 72.0? d. What statistics a student is selected at random, what is the probability they achieved a grade between 65.0 and $5.0? e, what statistics a student is selected at random, what is the probability they achieved a grade b een 80.0 and 90.07 OProblem 5 (20 Points). We have the following sample space: 2 - {1, 2, 3. 4). Ler the probability of singleton sets be as follows: for a constant B > 0. P({}) = 1B. for i = 1, 2. 3. 4. Thus, P((1}) = B. P({2}) = 2B. P({3}) = 3B, and P({4] ) = 4B. 1. Find the constant B so that the probability assignments satisfy the axioms of probability. 2. Find the probability that the outcome is odd. Find the probability of the ser {1, 2. 43.Problem 1 (20 points): Consider a sample space S consisting of a countable number of outcomes. Suppose that a probability function P has been defined that satisfies the three Axioms of Probability. Let E be an event with non-zero probability, so that P(E) > 0. For an arbitrary event F, consider the conditional probability defined as P(FIE) = P(FE)P(E). Prove that the conditional probability is also a probability. That is, prove that it satisfies the Axioms of Probability. Show all work. Prove every step unless it is obvious (even then a reason is recommended)