Favorite toy

The favorite toy is a formula created by Bill James that estimates the probability that a player achieves a cumulative statistical goal.

The formula has four inputs:

1) Number needed to achieve goal.

2) Projected seasons remaining for the player. This is calculated by the formula (24 - .6(age)). Any player still playing regularly has a minimum of 1.5 seasons remaining, regardless of age.

3) Established level for that statistic. James calculates this as (last season*3+second to last season*2+third to last season)/6. Additionally, the established level should not be below 3/4 of the total in the most recent season [see note below].

4) Projected remaining total, which is simply the last two numbers multiplied.

The chance of getting to the goal is calculated as:

[Projected remaining total - (Number needed/2) / Number needed]

There are two additional caveats:

a) A player can't have a likelihood greater than .97 ^ the estimated number of seasons it will take him to reach the goal.

b) A below average offensive player can't have a likelihood greater than .75 * the estimated number of seasons it will take him to reach the goal

note: Although James doesn't mention this specifically, it's unlikely that he intended the 3/4 rule to apply to high variance statistics such as no-hitters thrown by a pitcher

Sample Problem

Using this method, what is the chance that Alex Rodriguez hits 700 HRs? [Analysis as of December 8 2012]

Rodriguez is 37 years old. He has 647 career home runs. He was an above-average offensive player in the 2012 season.

He needs 53 more HRs to reach the goal.

His projected seasons remaining is (24 -(.6*37)) = 1.8

His current established HR level is (1*30) + (2*16) + (3*18) / 6 = 19.33

He's projected to hit (19.33*1.8) = 34.8 HRs

Therefore, by this method his chance of getting to 700 HRs is:

( 34.8 - (53/2) ) / 53 which equals .157 or 15.7%