A multiple-choice examination consists of 85 questions, each having possible choices a, b, c, and d. Approximate the probability that a student will get more than 22 answers correct if he randomly guesses at each answer. (Note that, if he randomly guesses at each answer, then the probability that he gets any one answer correct is 0.25.) Use the normal approximation to the binomial with a correction for continuity.Round your answer to at least three decimal places. Do not round any intermediate steps.

Find the indicated probability and interpret the result. From 1975 through 2020, the mean annual gain of the Dow Jones Industrial Average was 653. A random sample of 33 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 800? Assume σ = 1540. The probability is ☐ (Round to four decimal places as needed.)

A group of people were asked if they had run a red light in the last year. 147 responded "yes", and 399 responded "no".Find the probability that if a person is chosen at random, they have run a red light in the last year.  Give your answer as a fraction or decimal accurate to at least 3 decimal places

A medical test is available to determine whether a patient has a certain disease. To determine the accuracy of the test, a total of 9700 people are tested. Only 100 of these people have the disease, while the other 9600 are disease free. Of the disease-free people, 9300 get a negative result, and 300 get a positive result. The 100 people with the disease all get positive results. Round your answers to four decimal places, if necessary.   Find the probability that the test gives the correct result for a person who has the disease.