Table 4.25 Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing post-graduate research. He has the following probability distribution. x P(x) \"PM 1 0.35 2 0.20 3 0.15 4 5 0.10 5 0.05 Table 4.26 23. Define the random variable X. 24. Define P(x), or the probability of x. 25. Find the probability that a physics major will do post- graduate research for four years. P(x = 4) = 6 0.4 6(0.4) = 2.4 (6-5.4)2(0.4) = 0.144 8 0.2 8(0.2) = 1.6 (8-5.4)2(0.2) = 1.352 Table 4.23 21. Identify the mistake in the probability distribution table. X P(X) X*P(X) 0.15 0.15 2 0.25 0.50 3 0.30 0.90 4 0.20 0.80 5 0.15 0.75 Table 4.24 22. Identify the mistake in the probability distribution table. X P(x) X*P(X) 0.15 0.15 2 0.25 0.40 3 0.25 0.65 4 0.20 0.85Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time. 13. Define the random variable X. 14. What values does x take on? 15. Construct a PDF table. 16. Find the probability that Javier volunteers for less than three events each month. P(x 0) = 4.2 Mean or Expected Value and Standard Deviation 18. Complete the expected value table. 26. Flnd the probability that a physics major will do post- graduate research for at most three years. P(x 5 3) = 27. On average, how many years would you expect a physics major to spend doing post-graduate research? Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that she can plan what classes to offer. Over the years, she has established the following probability distribution. ' Let X = the number of years a student will study ballet with the teacher. ' Let Plx) = the probability that a student will study ballet x years. 28. Complete Table 4.27 using the data provided. X PlX) X*P(X) 1 0.10 2 0.05 3 0.10 4 5 0.30 29. In words, define the random variable X. 30. PO! :4) = 31.P(x3) = P(4 or 5 or 6) = 0.1 + 0.1 + 0.05 = 0.25 4. On average, how long would you expect a new hire to stay with the company? 5. What does the column \"P(x)" sum to? Use the following information to answer the next six exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution. X Table 4.20 #WM Ple 0.15 0.35 0.40 0.10 6. Define the random variable X. 7. What is the probability the baker will sell more than one batch? P(x >1): 8. What is the probability the baker will sell exactly one batch? P(x = 1): 9. On average, how many batches should the baker make? Use the following information to answer the next four exercises: Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random. 10. Define the random variable X. 11. Construct a probability distribution table for the data. 12. We know that for a probability distribution function to be discrete, it must have two characteristics. One is that the sum of the probabilities is one. What is the other characteristic