The Odds of Ichiro's Hot Streak: Fun Times with Binomial Probability

In the 5 games from September 19th to the 23rd, Ichiro had 14 hits in 20 ABs. What are the odds of that?

Binomial probability provides a way to calculate these odds. For the probability factor - the probability that Ichiro will get a hit in a given at-bat - we'll use batting average data. Since this probability is for an ongoing season, we'll use Ichiro's projected 2012 batting average of .288 as calculated by Marcels. Marcels projected he would hit .288.

Note: .288 is nearly identical to his actual 2012 batting average to date, which gives me confidence in using it.

With .288 as the probability factor, binomial probability indicates the odds of Ichiro's 14-for-20 streak is 1 out of 7226. For context, Ichiro has 8039 career ABs; meaning that if Ichiro were a .288 lifetime hitter, these 20 at-bats would have been a once-in-a-career event.

This made me curious about how often Ichiro went 14/20 in his prime years of 2001-2010. In that period, he hit .331. The binomial probability of a .331 hitter going 14-for-20 is 1 out of 1519; meaning the binomial probability estimates he would have had 4 14-20 streaks.

In reality in that period, Ichiro went 14-20 9 times; 7 times in August and September of 2004, and twice in June 2006.

9 is reasonably close to 4 or 5. Moreso perhaps when you consider that 4 of these 20-consecutive-at-bat streaks occurred in the August 14/15 games; they were part of a larger hot streak.

Finally, let's look at one more set of stats. The following table shows:

• The number of times Ichiro got a given number of hits in 20 ABs in the years 2001-2010;
• The number of times the binomial probability estimates it should happen. 

For example, Ichiro went 2-for-20 114 times; the binomial probability estimates it should happen 99 times. As you can see, the actual and estimated numbers are close (their correlation coefficient is .9987):

Hits/20 AB	Actual	Predicted
0	3	2
1	28	21
2	114	99
3	317	293
4	638	617
5	982	977
6	1213	1208
7	1144	1196
8	899	961
9	627	634
10	315	345
11	152	155
12	104	58
13	44	18
14	9	4
15	0	1
16	0	0
17	0	0
18	0	0
19	0	0
20	0	0