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

d. 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 long-term probability?