Hot Hands, a Paradox, and one reason why it’s bad to combine within-subject data

The ‘Hot Hand’ phenomenon is a popular belief (applicable to many domains from sports to gambling) that players who were successful in their most recent attempt(s) have increased odds of being successful in their next attempt — they are on a so-called ‘hot streak’ or have a ‘hot hand’. The statistical validity of this belief can be investigated using actual data. Indeed, it has been. For example, Tversky and Gilovich (1989) investigated the hot hand belief in basketball.

Here however, we are not scrutinizing the hot hand belief, but rather using this framework and dataset to reveal the presence of a Simpsonian Paradox. The definition of this paradox will precipitate from the following example…

In the 1996/97 NBA jam seasons Michael Jordan shot a pair of free throws on 338 occasions. MJ made both 251 times, missed both 5 times, made only the first 34 times, and made only the second 48 times. These data are presented in the table above, as are the same data for Dennis Rodman, and also their combined numbers.

Let Phit and Pmiss denote the proportion of first shot hits followed by a hit, and the proportion of first shot misses followed by a hit, respectively. These proportions for Jordan and Rodman, along with their combined numbers are:

Phit = 251 / 285 = .881
Pmiss = 48 / 53 = .906

Phit = 54 / 91 = .593
Pmiss = 49 / 80 = .612

Phit = 305 / 376 = .811
Pmiss = 97 / 133 = .729

Notice that, contrary to the hot hand phenomenon, MJ actually shot worse after he made the first of two freethrow shots. The same goes for Rodman.

So both players are actually worse on their second freethrow, on attempts when they’ve made their first shot.

Combining MJ and Rodman’s freethrow data together, the opposite is true. This is the Simpsonian paradox.

Should the sexual abuser come forward before the victim decides it’s time?

The recent flurry of sexual harassment and rape accusations naming a myriad of powerful men in Hollywood seems almost like a Zeitgeist. Since Harvey Weinstein exploded into public view there have been new allegations made against other powerful men in Hollywood and other industries almost daily. The courage of other victims coming forward has inspired and empowered Women with similar experiences to share their own story of sexual exploitation and injustice.

There’s no doubt, though, that many have yet to come forward. Each with their own personal reasons for not doing so, yet, or maybe ever.

And then there is the growing list of men to talk about. Some have denied, some have admitted, but none of them have released statements that suggested they genuinely sorry for their perversions. This was made more obvious after reading an actual apology; the one given by Louis C.K. just yesterday. I’m not saying a sincere apology atones for his actions, as he himself concedes in his letter. Just that it’s a good place to start, towards reform, and perhaps offering some redemption for the wronged.

My question though, isn’t about Louis C.K. and whether his apology was sufficiently conciliatory. My question is about whether it would be right for someone in his situation to address other instances where they engaged in sexual misconduct. That is, I feel like if someone is truly sorry for their previous actions, they should want to admit to all of them. Admit they’ve engaged in, not just a single lapse of judgement, but an entire pattern of bad behavior. Because from our perspective, for example, if another woman comes out tomorrow they were sexually assaulted by Louis C.K. he is back to square-one, as people are going to dismiss his apology as ingenuine.

On the other hand… what of the rights of the victimized? As I mentioned above, each person has their own reasons for not thrusting their traumatic experience into the public limelight.

With that in mind, see title. Discuss.

The Donald

A vain man childishly fishing for the adoration of strangers. Being rich isn’t enough; he wants to be universally loved for his imperfections, too. Slurping at the popularity spigot, The Donald’s need for ego gratification is double the size of any ordinary adult. Never in history has such a supercilious dope been more adored. It is beyond reason. Perhaps aware of this, and sensing defection, Trump whips out his phone… 140 characters is all it takes. Tweet.

He glances out the window of his private jet at the destruction beset a once beautiful US island; it disgusts him. He retreats to his phone to check emails, but years of operant conditioning and muscle memory take over. He taps on the bird icon. “30k likes! Only.“. Trump works best knowing his base is in a frothy outrage. “Better do one more…

That’s the one! Mmmmm. That one had everything.