In the game of​ roulette, a player can place a ​$ 4 bet on the number 27 and have a StartFraction 1 Over 38 EndFraction 1 38 probability of winning. If the metal ball lands on 27​, the player gets to keep the ​$ 4 paid to play the game and the player is awarded an additional ​$ 140 . ​ Otherwise, the player is awarded nothing and the casino takes the​ player's ​$ 4 . What is the expected value of the game to the​ player? If you played the game 1000​ times, how much would you expect to​ lose?

Please work problem step by step please thank you

An internet research company surveyed 90 online shoppers, each of whom made one purchase today. The company recorded the type of purchase each shopper made. Here is a summary. Type of purchase Number of shoppers Beauty supplies 12 Clothing 33 Pet supplies 21 Office supplies 24 Three shoppers from the survey are selected at random, one at a time without replacement. What is the probability that none of the three shoppers purchased beauty supplies? Do not round your intermediate computations. Round your final answer to three decimal places.

Suppose that two defective refrigerators have been included in a shipment of eight refrigerators. The buyer begins to test the eight refrigerators one at a time. Let the random variable Y represent the number of defective refrigerators found after three refrigerators have been tested. Compute the probability distribution of Y.

I did a similar problem to this but worked out a sample space etc. and solved for certain probabilities however what confused me then and what confuses me now is why is the sample space not 2 ^ (8) and rather 8 C 2

A random sample of 15 people was asked to record how much TV they watched every day for a week. The total number of hours each person watched during the week is displayed below.

The data are shown below.

b. If a normal approximation is appropriate, find the probability that a randomly selected person in the population watches less than 9 of TV per week. If a normal approximation is not appropriate, type “NA” (without the quotes).

I got the following equation from the lesson's summary:

P(X=k) = (e^-µ  µ^K) / k!

When calculating the probability while answering the homework problems I always seemed to be off by a very small amount. The only explanation given under the problem after hitting show answer is 1 - P(x=0) -P(x=1) - ... up until x = the number given in the question. I am confused where this equation or these types of calculations are coming from and would love some help!

Using R-studio

2. Consider an experiment where we flip a fair coin six times in a row, and i is the number of heads tossed:

            a.         Calculate the probability mass function for i = 0. . . 6 using the equation from Ross section 2.8 for Binomial Random Variables

            b.         Conduct a simulation of this experiment in R, with T trials of the experiment – pick several values of T from 10 to 10,000.

            c.         Create a plot of the theoretical result vs. your simulation at T = 100 and T = 10,000. Show that they converge as T increases.

BINOMIAL PROBABILITIES Big Box Store (BBS) has an annual rate of 4% of all sales being returned. In a recent sample of thirty randomly selected sales the number of returns was five. BBS is concerned about the event, and your advice is solicited. FOR 4 – 9, OPEN THE EXCEL EXAM TWO FILE. THE SPREADSHEET FOR THIS PROBLEM IS FOUND ON SHEET TWO. COMPLETE AND SAVE YOUR WORK IN EXCEL AND ENTER THE SOLUTIONS BELOW. (6) (2.5points) What is the probability that a random sample of 30 sales has 5 or fewer returns? Type the answer and any work below.  

1. John is going to take a quiz with 10 MC questions. Each question has only one correct answer out of 4 choices. Suppose John simply answer each question by random guess,
   1. (5) What is the probability of have 5 questions answered correctly?
   2. (5) What is the expected number and standard deviation of the questions answered correctly?
   3. (10) If a correct answer counts for 1 point and a wrong answer causes 1 point deduction,
      1. list out all the possible points of this quiz.
      2. what is the expected points of John’s quiz?

A) An Olympic archer is able to hit the bull’s eye 80% of the time. Assume each shot is independent of the others. She will shoot 6 arrows. Let X denote the number of bull’s eyes she makes.

Find the mean of the probability distribution of X. Do not round

B) The GPA of students at a college has a mean of 2.9 and a standard deviation of 0.3. Scores are approximately normally distributed.

Suppose that the top 6% of students are eligible for the Honors Program. Find the GPA which is the cutoff score for students to qualify for this program. Round to the nearest hundredth.