Modeling Rare Baseball Events — Are They Memoryless?

Figures & data

Figure 1. Number of No-Hit Games by Year (1901 – 2004).

Figure 2. Number of Cycles by Year (1901 – 2004).

Figure 3. Number of Triple Plays by Year (1901 – 2004).

Table 1. Actual (versus Estimated) Counts of Rare Events Per Season (1901–2004)

Count	No Hitters	Hit for Cycle	Triple Plays
0	18 (14.3)	18 (12.0)	2 (0.8)
1	30 (28.4)	27 (25.9)	6 (3.8)
2	21 (28.1)	16 (28.0)	12 (9.2)
3	21 (18.6)	21 (20.2)	12 (15.1)
4	6 (9.2)	13 (10.9)	17 (18.6)
5	3 (3.6)	5 (4.7)	18 (18.2)
6	3 (1.2)	3 (1.7)	9 (14.9)
7	2 (0.3)	0 (0.5)	11 (10.5)
8	0 (0.1)	1 (0.1)	5 (6.4)
9	0 (0.0)	0 (0.0)	7 (3.5)
10	0 (0.0)	0 (0.0)	2 (1.7)
11	0 (0.0)	0 (0.0)	3 (0.8)
12	0 (0.0)	0 (0.0)	0 (0.3)
Total	206	225	511
λ = x̄ s^2	1.98 2.64	2.16 2.90	4.91 6.66

Table 2. Estimated Probabilities of Rare Events Occurring Per Year Using the Poisson Distribution Model and Estimated Mean

#	No Hitters	Hit for Cycle	Triple Plays
0	13.80%	11.49%	0.73%
1	27.33%	24.86%	3.61%
2	27.06%	26.90%	8.87%
3	17.87%	19.40%	14.53%
4	8.85%	10.49%	17.84%
5	3.51%	4.54%	17.53%
6	1.16%	1.64%	14.36%
7	0.33%	0.51%	10.08%
8	0.08%	0.14%	6.19%
9	0.02%	0.03%	3.38%
10	0.00%	0.01%	1.66%
11	0.00%	0.00%	0.74%
12	0.00%	0.00%	0.30%
13	0.00%	0.00%	0.11%
14	0.00%	0.00%	0.04%
15	0.00%	0.00%	0.01%

Figure 4. EDF vs. Exponential CDF for No-Hit Games.

Table 3. The Construction of the Chi-square Test Statistic for No-Hitters

IAT		Observed (obs)	Expected (exp)	(obs – exp)^2/exp
≥	<
0	56	20	14.4085	2.1699
56	162	20	24.5716	0.8505
162	221	20	12.2843	4.8462
221	328	20	20.0194	0.0000
328	530	20	31.0046	3.9059
530	638	20	13.5348	3.0882
638	820	20	18.9327	0.0602
820	1132	20	23.6778	0.5713
1132	1654	20	23.3695	0.4858
1654	2762	20	18.4333	0.1332
2762	4029	6	4.6461	0.3945

Table 4. Historical Eras of Major League Baseball With Occurrences

Era	Years	No-Hitters	Cycles	Triple Plays
Dead Ball	1901 to 1919	43	22	107
Lively Ball	1920 to 1941	20	60	136
Integration	1942 to 1960	30	29	81
Expansion	1961 to 1976	56	31	66
Free Agency	1977 to 1993	37	46	81
Long Ball	1994 to 2004	20	37	40

Table 5. Mean Inter-Arrival Times and Lambda Values Per Era

		No-Hitter	No-Hitter	Cycles	Cycles	Triple Play	Triple Play
Era	Seasons	Mean IA	λ	Mean IA	λ	Mean IA	λ
Dead Ball	19	536	2.26	959	1.16	213	5.63
Lively Ball	22	1344	0.91	449	2.73	199	6.18
Integration	19	788	1.58	798	1.53	285	4.26
Expansion	16	514	3.50	899	1.94	415	4.13
Free Agency	17	889	2.18	747	2.71	444	4.76
Long Ball	11	1164	1.82	665	3.36	613	3.64

Table 6. Anderson-Darling Goodness-of-Fit Tests

Stat Era	Years	A^2	n	adj A^2	adjusted p-value	Exponential?
all triple plays	1901 – 2004	2.73	511	2.73	< 0.0025	Reject
Dead Ball	1901 – 1919	0.48	107	0.48	> 0.25	Do Not Reject
Lively Ball	1920 – 1941	0.67	136	0.68	> 0.25	Do Not Reject
Integration	1942 – 1960	0.35	81	0.35	> 0.25	Do Not Reject
Expansion	1961 – 1976	1.05	66	1.06	= 0.01	Reject
Free Agent	1977 – 1993	0.51	81	0.51	> 0.25	Do Not Reject
Long Ball	1993 – 2004	0.72	40	0.72	> 0.25	Do Not Reject
all Cycles	1901 – 2004	1.03	225	1.03	0.10 < p < 0.15	Do Not Reject
Dead Ball	1901 – 1919	2.96	22	3.04	< 0.0025	Reject
Lively Ball	1920 – 1941	0.94	60	0.95	0.10 < p < 0.15	Do Not Reject
Integration	1942 – 1960	0.72	29	0.73	> 0.25	Do Not Reject
Expansion	1961 – 1976	0.72	31	0.73	> 0.25	Do Not Reject
Free Agent	1977 – 1993	1.01	46	1.02	0.10 < p < 0.15	Do Not Reject
Long Ball	1993 – 2004	0.90	37	0.91	0.15 < p < 0.20	Do Not Reject
all No-Hitters	1901 – 2004	0.81	206	0.81	0.20 < p < 0.25	Do not Reject
Dead Ball	1901 – 1919	0.96	43	0.97	0.10 < p < 0.15	Do Not Reject
Lively Ball	1920 – 1941	1.43	20	1.47	0.025 < p < 0.05	Reject
Integration	1942 – 1960	0.95	30	0.97	0.10 < p < 0.15	Do Not Reject
Expansion	1961 – 1976	0.63	56	0.64	> 0.25	Do Not Reject
Free Agent	1977 – 1993	1.92	37	1.95	0.01 < p < 0.025	Reject
Long Ball	1993 – 2004	1.08	20	1.11	0.05 < p < 0.10	Do Not Reject