## 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.

### WHEN WAS JOHN TYLER BORN?

1. Consider the year of birth of John Tyler, former President of the United States. The table below lists four statements about Tyler's year of birth.

YEAR OF BIRTH PROBABILITY
no later than 1750
between 1751 and 1775
between 1776 and 1800
after 1800

• 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 information that John Tyler was the tenth president of the United States. Use this information to reevaluate the probabilities you made above. Before you assign probabilities, answer the first two questions stated above.

3. You are now given the information that George Washington, the first President of the United States, was born in 1732. Again reevaluate your probabilities and answer all three questions.