May 11, 2007

Counting Down to 756

When, and Against Who?

by Clay Davenport

Love him, hate him, or just don't care about him, Barry Bonds is on the cusp of eclipsing baseball's most cherished record, the title for most career home runs. Want to know when he'll do it?

Using essentially the same system as the playoff odds report, we can model Bonds' home run hitting on a game-by-game basis in order to get an estimate for just when #756 will carry over an outfield fence. All of this will assume that the chances remain relatively stable--if he deliberately avoids hitting home runs on the road (by, say, bunting against the shift), or if pitchers start walking him even more often to make sure they're not the next Al Downing, then all bets are off.

In its simplest form, we need to know

1. How likely is he to play in any given game?
2. How many plate appearances will he get in the games he plays?
3. How likely is he to hit a home run in each plate appearance?

Last year, Bonds played in 130 of his team's 161 games; so far this year, he's played in 31 of 34. The model is going to assume that he generally has a 90% chance of playing in any particular game, assuming he's not hurt--we'll get to that in a minute. However, given the stated preference for setting the record at home rather than on the road, as well as the ovation, reaction, and opprobrium he receives from the same split, we're going to bias the stats a little bit towards home games; rather than a flat 90%, let set it at 95% at home and 85% on the road. We'll also decree that he won't play both ends of their scheduled doubleheader in Pittsburgh on August 13, if he's still under the record.

Injuries, though, represent a real issue, one that has to be taken into consideration. Without them, there is a pretty much 100% chance that Bonds will break the record this year. I don't think it's nearly so clear that he'll stay healthy--we're talking about someone who missed virtually all of 2005, after all, who could do further damage to his knees at almost any time. To account for that, I built a 1% chance of injury into every game he plays, a percentage that is probably a little bit on the low side, given his history and status. You might expect that he'll probably play more cautiously than usual to try and avoid injuries. Any such injury in my system forces him out until he recovers; the recovery chance starts out at modest 20% chance of returning, but each day he misses reduces the chance of returning the day after that; think of it as saying that every game missed indicates the injury was a little more serious than first thought. The way it is set up, an average injury will cost Bonds10 games--but it could also last the rest of the season.

Once we establish that he plays in a particular game, the rest is more straightforward. Between last year and so far this season, Bonds has averaged 3.79 PA/game, so I simply adapted his real distribution of PAs per game for use here. It is heavily weighted towards 3 or 4 per game, but can be anything between 1 and 6.

For his home run rate, we'll lean on last year's results pretty heavily, with a nod towards this year, and we'll let that change as the year progresses. That means I'm going to combine his 2006 and 2007 performances and use the total to set his home run rate per PA. We'll adjust that depending on what park he's playing in, but won't worry about the opposing pitchers--the variability even within one pitching staff is just too large.

If we didn't allow any injuries in the model, then Bonds always reaches 756 this season. The distribution after a million runs looks like this (starting with games of May 11):

Earliest reach   May 17     @ Houston
10% chance by    June 3     @ Philadelphia
20%              June 6     @ Arizona
30%              June 9     Oakland
40%              June 11    Toronto
50%              June 15    @ Boston
60%              June 17    @ Boston
70%              June 20    @ Milwaukee
80%              June 24    Yankees
90%              June 30    Arizona
Latest           August 29  Colorado

The basic numbers put the most likely time period for 756 in the first three weeks of June, when the Giants are playing 13 of 19 on the road; I'm expecting some sort of nagging injury to dog him through then, with an eye towards the nine-game homestand against the Yankees, Padres, and Diamondbacks that closes out June into July.

Making an explicit injury allowance makes for some dramatic changes at the back end; the longer it runs, the more likely an injury is going to affect the outcome.

Earliest  reach      still May 17 in Houston
10% chance by        June 4, back one day
20% chance by        June 7, back one day
30% chance by        June 10, back one day
40% chance by        June 13, back two days
50% chance by        June 16, back one day
60% chance by        June 20, back three days
70% chance by        June 25, back five days
80% chance by        July 3, back nine days
90% chance by        August 9, back forty days

With an explicit injury provision, Bonds did not reach 756 home runs in 6.4% of the million seasons I ran, about one chance in 16, which gives those who hate him a meaningful rooting interest. Again, though, the peak likelihood corresponds to the six-game road trip to Boston and Milwaukee from June 15-20.

Actually, that's only the peak cumulative likelihood, which is distorted because the tail is so much longer than the front. The single most likely day, with a 3.75% chance, falls on June 11 (at "only" a 36% cumulative chance), with a 3% chance or better every day from June 6^th through the 19^th. The two most likely series to see him reach the milestone are both at home, led by the Toronto series from June 11-13 and the Oakland series of June 8-10, with an 11.2 and 10.8% chance, respectively. The next four most likely series are all on the road--10.5% in Boston June 15-17, 9.1% in Arizona June 5-7, 9.0% in Milwaukee June 18-20, and 8.0% in Philadelphia from June 1-4. The next three most likely come after the Giants return home from Milwaukee--7.2% against the Yankees from June 22-24, then 5.55% against the Padres on the 25-27, then 4.1% against the Diamondbacks from June 29 through the first of July. The overall odds are slightly tilted towards it happening on the road.

The complete list from this morning's run:

            This day  Cumulative
137 May 17         4       4            @HOU
138 May 18        18      22            @OAK
139 May 19        50      72
140 May 20       143     215
141 May 21       358     573            HOU
142 May 22       663    1236
143 May 23      1254    2490
145 May 25      2094    4584            COL
146 May 26      3244    7828
147 May 27      4920   12748
149 May 29      6962   19710            @NYM
150 May 30      9280   28990
151 May 31     12146   41136
152 Jun  1     15020   56156            @PHI
153 Jun  2     18669   74825
154 Jun  3     21378   96203
155 Jun  4     25015  121218
156 Jun  5     27813  149031            @ARI
157 Jun  6     30418  179449
158 Jun  7     32717  212166
159 Jun  8     34754  246920            OAK
160 Jun  9     35981  282901
161 Jun 10     36985  319886
162 Jun 11     37536  357422            TOR
163 Jun 12     37246  394668
164 Jun 13     36964  431632
166 Jun 15     35932  467564            @BOS
167 Jun 16     35107  502671
168 Jun 17     33547  536218
169 Jun 18     32030  568248            @MIL
170 Jun 19     30015  598263
171 Jun 20     28274  626537
173 Jun 22     26092  652629            NYY
174 Jun 23     24148  676777
175 Jun 24     22137  698914
176 Jun 25     20254  719168            SDP
177 Jun 26     18526  737694
178 Jun 27     16674  754368
180 Jun 29     15202  769570            ARI
181 Jun 30     13258  782828
182 Jul  1     12193  795021
184 Jul  3     10862  805883            @CIN
185 Jul  4      9707  815590
186 Jul  5      8583  824173
187 Jul  6      7749  831922            @STL
188 Jul  7      6846  838768
189 Jul  8      5961  844729
194 Jul 13      5441  850170            LAD
195 Jul 14      4751  854921
196 Jul 15      4325  859246
197 Jul 16      3784  863030            @CHC
198 Jul 17      3320  866350
199 Jul 18      2976  869326
200 Jul 19      2733  872059
201 Jul 20      2576  874635            @MIL
202 Jul 21      2325  876960
203 Jul 22      2033  878993
204 Jul 23      1900  880893            ATL
205 Jul 24      1825  882718
206 Jul 25      1757  884475
207 Jul 26      1518  885993
208 Jul 27      1444  887437            FLA
209 Jul 28      1348  888785
210 Jul 29      1286  890071
212 Jul 31      1219  891290            @LAD
213 Aug  1      1205  892495
214 Aug  2      1169  893664
215 Aug  3      1135  894799            @SD
216 Aug  4      1055  895854
217 Aug  5      1066  896920
218 Aug  6       996  897916            WAS
219 Aug  7       979  898895
220 Aug  8       962  899857
221 Aug  9       951  900808
222 Aug 10       946  901754            PIT
223 Aug 11       932  902686
224 Aug 12       920  903606
225 Aug 13       906  904512            @PIT (2)
226 Aug 14       926  905438            @ATL
227 Aug 15       882  906320
228 Aug 16       818  907138
229 Aug 17       913  908051            @FLA
230 Aug 18       827  908878
231 Aug 19       845  909723
232 Aug 20       849  910572
233 Aug 21       806  911378            CHC
234 Aug 22       800  912178
235 Aug 23       804  912982
236 Aug 24       829  913811            MIL
237 Aug 25       760  914571
238 Aug 26       816  915387
239 Aug 27       764  916151            COL
240 Aug 28       773  916924
241 Aug 29       753  917677
243 Aug 31       734  918411            @WAS
244 Sep  1       754  919165
245 Sep  2       760  919925
246 Sep  3       752  920677            @COL
247 Sep  4       762  921439
248 Sep  5       710  922149
250 Sep  7       728  922877            LAD
251 Sep  8       667  923544
252 Sep  9       693  924237
253 Sep 10       704  924941            ARI
254 Sep 11       722  925663
255 Sep 12       650  926313
257 Sep 14       674  926987            @SD
258 Sep 15       655  927642
259 Sep 16       698  928340
260 Sep 17       646  928986            @ARI
261 Sep 18       630  929616
262 Sep 19       633  930249
263 Sep 20       688  930937            CIN
264 Sep 21       642  931579
265 Sep 22       639  932218
266 Sep 23       617  932835
267 Sep 24       608  933443            SD
268 Sep 25       572  934015
269 Sep 26       574  934589
271 Sep 28       616  935205            @LAD
272 Sep 29       618  935823
273 Sep 30       566  936389
Not this year  63611 1000000