A sixteen-sided number cube has the numbers 1 through 16 on each face. Each face is equally likely to show after a roll. What is the probability that you will roll an even One-hundred students were allowed to re-take an exam for their math course. The probability distribution shows how studying for the latest exam affected their grade when compared with the first time they took the exam. What is the probability that a student who studied for the exam saw an increase in their exam grade? Round to the nearest thousandth. number or an odd prime number? Round to the nearest thousandth.

Probaility in general is defined as the ratio of positive outcomes over the total number of outcomes.

In the first example, the total outcomes are 16; let us count the positive ones. There are 8 even numbers from 1-16. The prime numbers are 2,3,5,7,11,13. Out of those, only 5 are odd. Hence, in total there are 13 positive outcomes. Thus, the probability is 13/16=81.25%

Let's restrict the problem to the students that studied for the exam; the proportion is 0.57 of the total students. 0.52 of the total students studied and saw an increase in their exam. Hence, the probability that a student who studied saw an increse is 0.52/0.57 (here a positive outcome is the proportion that saw an increase and the total outcomes are all the students that studied). 0.52/0.57=91.22%