Discussion Board 6 Expected Value, Binomial & Poisson New applicants that sign up for a brokerage account will get 1 share of stock from the company. They are randomly assigned, use the information from the table to answer the following questions: Chris OByrne 2024x P(x)$4 0.23$9 0.34$15 0.20$20 0.13$35 0.06$60 0.04A. What is the expected value of the stock that a new applicant will receive (no rounding)?B. What is the probability that an applicant will get a stock valued at more than $9? Of all the civil lawsuits filed in court, only 6.4% of them actually go to trial. If you were to randomly select 12 civil lawsuits, use this information to answer the following questions (round final answer to 5 decimal places): Chris OByrne 2024A. What is the probability that more than 1 lawsuit will go to trial?B. What is the probability that at most 2 lawsuits will go to trial?C. What is the probability that the first civil lawsuit will go to trial on the 7th selection?D. What is the expected number of civil lawsuits that will go to trial?Chris OByrne 2024 Adults exercise 3.2 times per week. The probability that an adult exercises during the week is independent of whether they exercised or didnt exercise during anytime that week. Exercising happens randomly throughout the period and the probability they exercise is the same for any interval. If they exercise or dont exercise this does not influence or affect if they will exercise or not exercise in any other interval. Exercising happens randomly throughout the interval (round final answer to 5 decimal places): Chris OByrne 2024A. What is the probability that an adult will exercise at most 3 times during the week?B. What is the probability that an adult will exercise at least 2 times during the week?C. What is the probability that an adult will exercise 3 times during a 2-week period? (you need to adjust the period)Chris OByrne 2024 Share on Facebook Tweet Follow us Sample Answer Essay on Expected Value, Binomial Distribution, and Independent Events in Various Scenarios Introduction Probability plays a significant role in decision-making and understanding the likelihood of events occurring in different contexts. In this essay, we will explore expected value calculations, binomial distributions, and probabilities of independent events in scenarios involving brokerage account stock values, civil lawsuits going to trial, and adult exercise habits. Expected Value of Stock for New Brokerage Account Applicants When new applicants sign up for a brokerage account, they receive shares of stock randomly assigned from a distribution. Given the probabilities associated with different stock values, we can determine the expected value of the stock an applicant will receive. A. Expected Value Calculation The expected value is calculated by summing the products of each stock value and its corresponding probability: [ text{Expected Value} = 4 times 0.23 + 9 times 0.34 + 15 times 0.20 + 20 times 0.13 + 35 times 0.06 + 60 times 0.04 ] B. Probability of Receiving Stock Valued More Than $9 To find the probability that an applicant will receive a stock valued at more than $9, we sum the probabilities for $15, $20, $35, and $60: [ P(text{Stock Value} > $9) = P($15) + P($20) + P($35) + P($60) ] Probability of Civil Lawsuits Going to Trial In the realm of civil lawsuits, a small percentage actually proceed to trial. By considering the proportion of cases that go to trial, we can analyze various probabilities related to randomly selected civil lawsuits. A. Probability of More Than 1 Lawsuit Going to Trial Using the binomial distribution formula, we calculate the probability of more than one lawsuit going to trial out of 12 selected cases. B. Probability of At Most 2 Lawsuits Going to Trial Similarly, we employ the binomial distribution to find the probability of at most two lawsuits going to trial. C. Probability of First Trial on the 7th Selection Given that only 6.4% of cases go to trial, we determine the probability that the first civil lawsuit will go to trial on the 7th selection out of 12. D. Expected Number of Civil Lawsuits Going to Trial The expected number is computed by multiplying the total number of cases by the probability of a case going to trial. Probability of Adult Exercise Habits Understanding adult exercise patterns involves analyzing probabilities of exercising a certain number of times during a week or over a period. These probabilities are influenced by independent events and random occurrences. A. Probability of Exercising at Most 3 Times in a Week By considering the average exercise frequency and independence of events, we calculate the probability that an adult will exercise at most three times during a week. B. Probability of Exercising at Least 2 Times in a Week Applying similar principles, we determine the probability that an adult will exercise at least twice during a week. C. Probability of Exercising 3 Times in a 2-Week Period When extending the analysis to a two-week period, we adjust the probabilities to find the likelihood of an adult exercising exactly three times. Conclusion Probability distributions, binomial calculations, and considerations of independent events are essential tools in analyzing various scenarios such as stock allocations, legal proceedings, and health habits. By applying these concepts effectively, we can make informed decisions and draw insights into the likelihood of specific outcomes occurring in diverse situations. 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