A high school guidance counselor wants to find the probability that students who study geometry as freshmen will go on to study calculus before graduating. To do this, he has constructed a table, shown below, which shows the number of students who study geometry as freshmen in a given year, and the number of those students who study calculus before graduating. Year 2000 2001 2002 2003 2004 2005 Geometry Students 680 577 822 743 859 804 Calculus Students 391 336 512 374 465 361 Which year had the highest experimental probability that a freshman studying geometry would take a calculus class before graduation? a. 2001 b. 2002 c. 2003 d. 2004

To find the year with the highest experimental probability that a freshman studying geometry would take a calculus class before graduation, we need to calculate the probability for each year by dividing the number of students who took calculus before graduating by the total number of students who studied geometry as freshmen.

Here is the probability calculation for each year:

- For the year 2000: Probability = number of calculus students / number of geometry students = 391 / 680 Probability = 0.575 (rounded off to three decimal places)

- For the year 2001: Probability = number of calculus students / number of geometry students = 336 / 577 Probability = 0.582

- For the year 2002: Probability = number of calculus students / number of geometry students = 512 / 822 Probability = 0.623

- For the year 2003: Probability = number of calculus students / number of geometry students = 374 / 743 Probability = 0.503

- For the year 2004: Probability = number of calculus students / number of geometry students = 465 / 859 Probability = 0.541

- For the year 2005: Probability = number of calculus students / number of geometry students = 361 / 804 Probability = 0.449

Now we compare these probabilities to determine which is the highest. From the calculations, it is clear that the highest probability was in the year 2002, with a probability of 0.623.

The answer is b. 2002.

Extra: The concept of probability measures the likelihood of an event happening and is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In the context of this problem, the event we are considering is a student who studied geometry as a freshman going on to study calculus before graduating. The number of favorable outcomes is the number of students who take calculus, and the total number of outcomes is the number of students who took geometry.

Probability ranges from 0 to 1, where 0 means the event will not happen and 1 means that it is certain to happen. A higher probability value means an event is more likely to occur.

When calculating probabilities based on historical or experimental data, such as in this classroom example, it is known as experimental or empirical probability. This contrasts with theoretical probability, which would be based on mathematical models without direct reference to observed outcomes. Understanding probabilities is important in many fields, from science and engineering to economics, as it enables the prediction and analysis of events in uncertain conditions.