Short-run proportions are often quite different from long-run probabilities.

a. In your own words, explain why we would expect proportions to fluctuate in the short run, but why long-run probabilities are more predictable.

b. What is the expected long-run probability of heads if a person flips a coin many, many times? Why?

c. Flip a coin 10 times in a row. What proportion is heads? Do this 5 times. Note: You will learn more by actually doing it, so don’t just write down numbers!

Proportion for the first 10 flips:

Proportion for the second 10 flips:

Proportion for the third 10 flips:

Proportion for the fourth 10 flips:

Proportion for the fifth 10 flips:

d. Do the proportions in part (c) match the expected long-run probability in part (b)? Why or why not?

e. Imagine that a friend flipped a coin 10 times, got 9 out of 10 heads, and complained that the coin was biased. How would you explain to your friend the difference between short-term and longterm probability?