Is Babe Ruth’s statement meaningful?

Quote from Babe Ruth:

Every strike brings me closer to the next home run.

As I understand memorylessness, this is meaningless. For every at-bat, there is a certain probability that he will strike, and there is a certain probability that he will hit a home run, and that’s that. The likelihood of a home run at any particular point in time does not increase as strikes accrue.

However, I have an intuitive understanding of what he means. Is there some statistically-rigorous way to express it or make sense of it?

Maybe it makes sense for someone looking back on Babe Ruth’s career with the benefit of hindsight. Or, maybe if we imagine an omniscient deity who can see the entire timeline of the universe at once. The deity can indeed see that, from any particular moment, there are N strikes remaining before Ruth hits the next home run. Another strike reduces that number to N-1. So, indeed, every strike brings him closer to the next home run.

Epilogue

If I could go back in time and rewrite this question, I would have omitted all the baseball references and simply described a guy rolling dice, hoping for a seven. He says, “I’m hoping for a seven, but I’m not bothered when I get something else, because every roll of the dice brings me closer to that seven!” Assuming he eventually rolls a seven, is his assertion the gambler’s fallacy? Why or why not?

Thanks to @Ben for articulating that this is not the gambler’s fallacy. It would have been the gambler’s fallacy if he had instead said, “Every roll of the dice which does not result in a seven makes it more likely that the next roll results in a seven.”

The guy didn’t make any such statement, and he didn’t make any statement at all about probability, merely about the passage of time.

By assuming that there is a seven in his future, we have made it undeniably true that every roll of the dice brings him closer to the seven. In fact, it is trivially true. Every second that ticks by, even when he is sleeping, brings him closer to that seven.

Answer

It is both meaningful and (usually) correct

You are overcomplicating this by bringing probability into a simple non-probabilistic assertion. You need not invoke an omniscient deity in order to accept that there is a reality that exists independently of knowledge of it. (You seem to be operating under the assumption that reality is only admissible to discussion if there is an omniscient being with total knowledge of it; this is a reasonably common misconception of probability, which is examined in this related question.)

The simplest rigorous examination of this statement is a non-statistical analysis based on looking at the underlying population of values pertaining to all the balls Babe Ruth ever hit. Let X1,...,XN be the ordered career outcomes of all balls faced by Babe Ruth, with Xi= denoting a strike and Xi= denoting a home-run (we need not specify the notation for other possible outcomes). At the end of ball n the number of balls until the next home-run is:

Bnmin

Now, we know that a strike and a home-run are mutually exclusive — i.e., no single ball can be both. Consequently, if ball n+1 is a strike (i.e., if X_{n+1} = \bullet) and if B_n<\infty (i.e., if Babe has at least one home-run left in his career) then we can easily show that B_{n+1} = B_n-1. This confirms Babe's statement that his strike brings him (one ball) closer to his next home-run.

The only exception to this is when Babe gets to the point where he has already hit his last home-run, so that there are no more home-runs left to come in his career. At this point with have B_n = \infty and getting a strike on ball n+1 still gives B_{n+1} = \infty. In this latter case Babe is no closer to the next home-run, because there is no next home-run.

Of course, at the time of Babe's last home-run, he probably didn't know that would be his last. (According to this historical account, Babe's last home-run was on 25 March 1935. He went on to play five more times without another home-run.) At that point his saying would be wrong, and looking back in hindsight we now know this.

Ultimately, this statement by Babe Ruth is no more controversial than if he asserted, "The elapsing of time spent not getting a home-run brings me closer to my next home-run". That is of course also true, setting aside the situation where he has no future home-runs to get closer to.

Finally, I do not agree with other comments/answers here that assert that this is the gambler's fallacy. It could (but might not) be a manifestation of the gambler's fallacy if he instead said, "Every strike makes it more likely that I will get a home-run in the future". That could be an example of the gambler's fallacy because it would assert that a bad outcome now makes a good outcome in the future more likely. (On the other hand, if strikes are not independent then it might not be.) In any case, merely asserting that the elapsing of time required for a bad outcome to occur now makes a subsequent good outcome closer in time is not the gambler's fallacy, and is not a fallacy at all.

Attribution
Source : Link , Question Author : SlowMagic , Answer Author : Ben

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