## What’s the probability that Daniel Murphy hits a home run tonight?

20% 15%, that’s my quick empirical estimate.

Where do I get this? I googled *most home runs hit in consecutive games* and found this list of players who’ve hit home runs in at least six consecutive games. There are 20 such cases; 14 of these streaks ended at six games, and 6 of these streaks continued. So I’d estimate Pr(homer in 7th consecutive game | homered in 6 consecutive games) as 6/20 = 0.30.

Ok, but Murphy’s streak so far is only 5 games, not 6. So we really want Pr(homer in 6th consecutive game | homered in 5 consecutive games). How many streaks of length 5? According to this page, there have been 39 streaks of 5 or longer since 1997. Going back to that earlier page, we see that 7 of the streaks of length 6+ happened since 1997. So this gives us an empirical probability of 7/39 = 0.18.

So my empirical estimate is 18%, which is hyper-precise. I round it to 20%.

P.S. See discussion in comments. After partial pooling to Murphy-specific data, my new estimate is 15%.

P.P.S. My commenters know much more baseball than I do and are trying to push my probability down to 10%. I’ll let all of you make the call on this one.

P.P.P.S. The results of a more precise calculation appear in the next post.

1. Guy says:

The probability is considerably lower than 18%. You haven’t taken account of Murphy’s HR-hitting talent level. Just eyeballing your list of 39 streaks, at least 25 of those were produced by players with demonstrated HR-hitting ability (30+ HR/season). Nearly all of them hit HRs at a higher rate than Murphy.

You’d get a much better estimate by starting from Murphy’s HR/PA rate, which is 1.7% over his career. That would suggest about an 8% chance of hitting a HR.

• Andrew says:

Guy:

Good point, I guess we should partially pool, so let’s compromise: (.08 + .18)/2 = .13, which I’ll round to 15%.

2. J. Cross says:

Yeah, ditto what Guy said

Even taking Murphy’s performance including the playoffs at face value (his HR rates from previous years are lower) as his current skill level, we’re only getting to about a 3.5% chance of hitting a HR in any plate appearance. For 4 plate appearances, that means an 13.3% chance of a HR and for 5 plate appearances a 16.3% chance. I’m not sure what we know at this point about the wind at Wrigley field. Murphy does have the benefit of facing Hammel, a right-handed fly ball pitcher, to start the game.

SaberSim has Murphy projected to get a mean of 4.5 PA and a mean of only 0.089 HR implying only a 2% chance of hitting a HR in any plate appearnces. That estimate is based on projections that include Murphy’s career data, the ballpark and the pitcher but don’t include Murphy’s post-season performance or any hot-hand effect. The SaberSim projection can be found here: https://www.sabersim.com/game/gid_2015_10_21_nynmlb_chnmlb_1

• J. Cross says:

Note: The first sentence should read “…performance this season including the playoffs…”

3. Guy says:

Why would we “compromise” between one estimate based on this player’s demonstrated ability to hit HRs, and another based on the success of *much better* power hitters? And then round *up* from there?

Leaving aside the possibility of a strong wind blowing out in Wrigley tonight, our best estimate of Murphy’s HR rate is 2.0% (conveniently, B-Ref just put up their 2016 projections). That suggests a 9% chance of a HR tonight. If you want to bump that to 10% for the “hot hand” I won’t argue with that, but 15% is still far too high.

• Andrew says:

Guy:

Fair enough. And of course the hot hand goes both ways, in that there’s nothing stopping the Cubs from pitching around Murphy. In any case, my goal here is transparency rather than completeness; I thought it would be fun to do the quick calculation, and I very much appreciate the feedback in comments.

4. Guy says:

BTW, an interesting question is whether your historical data implies any hot hand effect was at work at the end of the 39 5-game streaks. I would guess that an estimate of those players’ collective true talent HR rate (at that point in their respective careers) would be something like 4.5%. If so, there would be a 19% chance of a HR in the next game — about what we observe. So unless I’ve badly overestimated the talent level of these players, no indication here of a hot hand.

5. Hank says:

Chance = 100%. Because Cubs.

6. Alex says:

What happened to the fancy titles? “Daniel Murphy’s chances of hitting a home run aren’t as high as you think…. and they’re dropping by the edit!”

• Sean S says:

For the title: “The one weird trick that explains Daniel Murphy’s chance of hitting a home run tonight against the Cubs.”

7. Jonathan (another one) says:

0 percent. It’s impossible.

[There’s a really good chance that this can’t be disproven… Plus, I hate the Mets.]

• Jonathan (another one) says:

A colleague suggested that the probability of getting the probability correct is 0, because of Murphy’s Law.

8. Chris G says:

1) As a former baseball player (high school) I think there is something to the “hot bat”. There are some days or weeks where you just see the ball better and your swing is better and there others where you couldn’t hit the ball to save your life.

2) Five straight games with a HR is nuts. That’s way beyond anything I’d associate with a hot bat effect.

3) I say Murphy’s performance at the plate is a non-stationary process. Averaging Guy’s estimate with the empirical probability seems reasonable enough.

4) So if it’s a 15% chance that he hits a HR then if I bet someone a beer that he does and he does then my return (rounding to the nearest whole beer) is a six-pack. I think I’d take that bet.

• Rahul says:

Re your #1. That almost seems axiomatic. Wouldn’t disagreeing with #1 imply one thinks a player is absolutely uniform in performance with no variance?

Doesn’t matter whether it concerns baseball or chess, are we disputing that players have some streaks of high performance?

• Chris G says:

> Wouldnâ€™t disagreeing with #1 imply one thinks a player is absolutely uniform in performance with no variance?

No, I don’t think so. One could argue – which I don’t – that fluctuations in performance are random about the mean, i.e., uncorrelated with the prior day’s or game’s performance. That would give some variance. I believe that there are periods of zero correlation but that there are also periods of positive correlation – and I suppose periods of negative correlation too although I can’t think of any personal experiences which I identify as such.

• Guy says:

Players clearly perceive themselves to have a “hot bat” at times. But there is essentially no evidence for it in MLB performance data. Here’s one simple test: look at Andrew’s list of 39 players who hit HR in 5 consecutive games. Tally their HR rates (HR/PA) in the season of their streak (each name is linked to their career stats), and use that to estimate the probability they would hit a HR in the next game assuming 4.5 PA. I think you will find the result is extremely close to what actually happened (18%).

And if hitting a HR in 5 consecutive games isn’t “hot,” then one has to wonder what would qualify.

9. skyjo says:

He just did!