To solve this problem, we need to find two different probabilities and then multiply them to get our final answer.
The first probability we need is the probability that the first ruler the teacher hands out has centimeter labels. Since there are 15 rulers with labels and the total number of rulers is 25 (15 with labels plus 10 without labels), we can find this by dividing the number of rulers with labels by the total number of rulers, thus getting 15/25 = 0.6. So, the probability that the first ruler has labels is 0.6.
For the second probability, we need to take into consideration that the teacher has already handed out one ruler with labels. This means that now, there are 14 rulers with labels and 10 rulers without labels, making a total of 24 rulers. So, the probability that the second ruler she hands out will NOT have labels is found by dividing the number of rulers without labels by the total number of rulers left, thus resulting in 10/24 = 0.4166666666666667.
Finally, the probability that the first ruler handed out has labels and the second ruler does not have labels is the product of the two probabilities we just found. So, we multiply 0.6 (the probability that the first ruler has labels) by 0.4166666666666667 (the probability that the second ruler does not have labels), and we get 0.25.
Therefore, the probability that the first ruler the teacher hands out will have centimeter labels and the second ruler will NOT have labels is 0.25.