## ACTIVITY - CHANGING PROBABILITIES WHEN NEW INFORMATION IS AVAILABLE (from DeGroot, 1970)

The point of this activity is to demonstrate that probabilities that we assign are conditional on our current knowledge. As we learn and gain more information, our probabilities about statements can change. Later, we will introduce a formula, Bayes' rule, which will tell us how to compute new probabilities when we get information.

### HOW LARGE IS PENNSYLVANIA?

1. Consider the area, in square miles, of Pennsylvania. The table below lists four statements about the state's size.

PERIOD PROBABILITY
less than 5,000 square miles
between 5,000 and 50,000 square miles
between 50,000 and 100,000 square miles
over 100,000 square miles

• Which of the four statements do you believe is most likely?

• Which of the statements do you believe is least likely?

• Give probabilities to the four events that are consistent with the answers you made above. Put your answers in a probability table like shown above.

2. You are now given the following information: The area of Alaska, the largest of the fifty states, is 586,400 sq. miles, and the area of Rhode Island, the smallest state, is 1,214 sq. miles. Use this information to reevaluate the probabilities you made above. Before you assign probabilities, answer the first two questions.

3. You are now given the information that when area is considered, Pennsylvania is the thirty-third largest of the 50 states. Again reevaluate your probabilities and answer all three questions.

4. You are now given the information that the area of New York, the 30th largest state, is 49,576 sq. miles. Answer the same three questions.

Page Author: Jim Albert (© 1996)
albert@bayes.bgsu.edu
Document: http://www-math.bgsu.edu/~albert/m115/probability/penn4.html