In basketball, when the probability of making a free throw is 0.50 and successive shots are independent, the probability distribution of the longest streak of shots made has μ = 4 for 25 shots, μ = 5 for 50 shots, μ = 6 for 100 shots, and μ = 7 for 200 shots.
a. How does the mean change for each doubling of the number of shots taken? Interpret.
b. What would you expect for the longest number of consecutive shots made in a sequence of (i) 400 shots and (ii) 3200 shots?
c. For a long sequence of shots, the probability distribution of the longest streak is approximately bell shaped and σ equals approximately 1.9, no matter how long the sequence (Schilling, 1990). Explain why the longest number of consecutive shots made has more than a 95% chance of falling within about 4 of its mean, whether we consider 400 shots, 3200 shots, or 1 million shots.