For all calculations, show how you found your answer. This could be simply showing what you typed in the calculator to get your answer. 1. A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of returning $1,000,000 profit, and a 40% chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 70% of no profit or loss, and a 20% chance of losing the million dollars. a. Construct a PDF for investment B and for investment C. Let the random variables A, B, and C represent the profit on the investment. Let P(A), P(B), and P(C) represent the probability of achieving that profit. Investment A has been done for you. Investment A (software): Profit A 5,000,000 1,000,000 -1,000,000 Probability P(A) 0.10 0.30 0.60 Investment B (hardware): Profit B Probability P(B) Investment C (biotech): Profit C Probability P(C) b. Find the expected value of each investment. Show work using the formula OR by showing what you put in your calculator to find the answer. c. Which is the safest investment? Defend your answer using statistics. d. Which is the riskiest investment? Defend your answer using statistics.2. At a charity event, 500 raffle tickets are sold for the chance to win prizes. Tickets are purchased for $5 each. The grand prize is a gift card worth $500. One first-place winner will win a $50 gift card, and two second-place winners will win $25 gift cards. Assume all raffle tickets are sold. a. Fill in the table to complete the PDF. b. From the perspective of a player purchasing a raffle ticket, find the expected value of the game. Show work for the expected value by showing what you typed in your calculator to get your answer. C. Interpret this expected value in a sentence. Outcome Grand prize First Place Second Place You Lose Net Gain in Dollars, X Probability, P(X)Calculate the probabilities for the binomial experiment. You may use the binomedf and binompdf calculator features but show what you typed in the calculator to get your answer. 3. Suppose a student takes a multiple-choice test with 25 questions, each with five possible answers, only one of which is correct. Assume this is a nightmare scenario where students are randomly guessing and know nothing about the content of the test. Round all probabilities to the nearest ten-thousandth (four decimal places). a. Find the probability that a student guesses the correct answer on exactly six questions. b. Find the probability that a student guesses the correct answer on less than 3 questions. c. Find the probability that a student guesses the correct answer on more than 7 questions. d. Find the probability that a student guesses correctly between 3 and 7 times (inclusive). e. Based on these probabilities, in a class of 35 students, about how many students would score more than 7 correct just by randomly guessing? f. Find the mean and standard deviation of this distribution. Show work using formulas. g. Write a sentence to interpret the mean. h. Use the rule that most outcomes fall within two standard deviations of the mean. What would be considered unusual outcomes in this experiment?In the final part of this worksheet, you will analyze a real situation, compare the situation to binomial probability, and determine if the researchers in the story have successfully proven their ideas. Read and follow directions very carefully! Inspired by: Shermer, M. (2002). Why People Believe Weird Things: Pseudoscience, superstition, and other confusions of our time. Holt Paperbacks. The Association for Research and Enlightenment, located in Virginia, conducts experiments to attempt to establish support for the idea of ESP. In one experiment, two people are separated by a screen. Between them is a machine with 5 buttons, each with a different symbol (plus sign, square, star, circle, or waves). The "sender" pushes one of the buttons and attempts to send psychic messages to the "receiver," who attempts to guess which button was pushed. They do this 25 times. + O The experimenter makes four claims about psychic powers of mind reading (ESP or "extra- sensory perception") based on their results in the experiment. He makes the following claims defining psychic ability: A score of 5 correct is average Chance is between 3 and 7 correct (inclusive) . Subjects in the experiment who guessed correctly less than 3 times have "negative ESP" (he never explains what this means) Subjects in the experiment who guessed correctly more than 7 times have psychic powers of ESP. You will use mathematics to evaluate whether his claims of psychic ability are correct. 1. Assume that sender and receiver are not psychic. That is, assume they are randomly guessing symbols. This is a binomial experiment X-B(25, 0.20). Let success = guessing the symbol correctly The number of trials is n = 25 There are five symbols. The probability of success for each trial is p = 0.20 2. The researcher says that scoring between 3 and 7 correct is "chance." He also says you can prove you have psychic powers of mind reading by getting more than 7 correct. Compare this to your calculations in problem 3 of this worksheet. Is the researcher's standard of proof for what is considered "psychic" valid? Defend your answer using statistics.The graph below represents a relative frequency histogram for this theoretical probability distribution. Theoretical Binomial Probabilities for n=25 and p=0.20 0.2 0.2 0.15 0.1 0.09 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Number of Successes Ms. Boring has performed this experiment with her students in face-to-face classes every semester since spring 2017. The results are in the graph below. Results of Psychic Test Spring 2017 Through Fall 2021 (n= 342) 03 0.25 0.2 Relative Frequency 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Number Correct 3. Compare the theoretical distribution to the observed distribution from student experiments. What do you observe? 4. Have Ms. Boring's students succeeded in proving they have psychic ability? Defend your answer using statistics.Experimenter's results: The researcher at the centers for Research and Enlightenment did this experiment with 35 subjects. Each subject performed the experiment twice. The first time they ran this experiment: Twenty-nine people guessed correctly between three and seven times. . Three people guessed the symbol correctly 2 times Three people guessed correctly $ times. The second time they ran this experiment (with the same people): Thirty-one people guessed correctly between three and seven times. One person guessed correctly 1 time . Two people guessed correctly 2 times . One person guessed correctly 9 times. Additionally: . The people who guessed correctly more than seven times were different in each experiment. . The people who guessed correctly less than three times were different in each experiment. 5. Compare the results that the researcher collected (summarized above) with the results of your calculations in problem 3 on this worksheet. What do you observe? 6. Did the experimenter from the Centers for Research and Enlightenment succeed in proving that the subjects who scored more than 7 correct were psychic? Defend your answer using statistics