Applicants have a 0.26 probability of passing a test when they take it for the first time, and if they pass it they can move on to the next stage. However, if they fail the test the first time, they must take the test a second time, and when applicants take the test for the second time there is a 0.43 chance that they will pass and be allowed to move on to the next stage. Applicants are rejected if the test is failed on the second attempt. $\mathbf{(a)}$ What is the probability that an applicant moves on to the next stage but needs two attempts at the test? $\mathbf{(b)}$ What is the probability that an applicant moves on to the next stage? $\mathbf{(c)}$ If an applicant moves on to the next stage, what is the probability that he or she passed the test on the first attempt?

In a certain chemical reaction, a substance is converted into another substance at a rate proportional to the square of the amount of the first substance present at any time t. Initially (t = 0) 50 g of the first substance was present; 1 hr later, only 10 g of it remained. Find an expression that gives the amount of the first substance present at any time t. What is the amount present after 2 hr?

A business tax form is either filed on time or late, is either from a small or a large business, and is either accurate or inaccurate. There is an $11\%$ probability that a form is from a small business and is accurate and on time. There is a $13\%$ probability that a form is from a small business and is accurate but is late. There is a $15\%$ probability that a form is from a small business and is on time. There is a $21\%$ probability that a form is from a small business and is inaccurate and is late. $\mathbf{(a)}$ If a form is from a small business and is accurate, what is the probability that it was filed on time? $\mathbf{(b)}$ What is the probability that a form is from a large business?