A study for several professional sports of the effect of a player’s draft position d (d = 1, 2, 3,...) of selection from the pool of potential players in a given year on the probability π of eventually being named an all star used the model logit(π) = a + β log d.

a. Show that π/(1 – π) = e^α d^β. Show that e^α = odds for the first draft pick.

b. In the United States, Berry reported α̂ = 2.3 and β̂ = – 1.1 for pro basketball and α̂ = 0.7 and β̂ = –0.6 for pro baseball. This suggests that in basketball a first draft pick is more crucial and picks with high d are relatively less likely to be all-stars. Explain why.