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?
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.
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.
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.