# May 2021

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## Optimal Putting in Disc Golf

I often have to make a decision when putting at the edge of the green during disc golf.  Should I try to put the disc in the basket (choice 1) or should I just try to put the disc near the basket (choice 2)? If I try to put it in the basket and I miss, sometimes the disc will fly or roll so far way that I miss the next shot.

In order to answer this question, I created a simplified model.  Assume:

• If I don’t try to put the disc in the basket (choice 2), I will land near the basket and get it in the basket on the next throw.
• If I do try to put it in the basket (choice 1), I will succeed with probability $p$, where $p$ depends only on distance to the basket.
• If I do try to put it in the basket and fail to get it in the basket (choice 1), the probability that I will succeed on the second try is $q$ where $q$ is constant which does not depend on distance.
• If I do try to put it in the basket (choice 1), fail to get it in the basket, and then fail again on my second try, then I will always succeed on the third try.

Using these assumptions, I can compute the average number of throws for each choice.

For choice 2, I will always use two throws to get the disc in the basket.

For choice 1, there are three possible outcomes:

• outcome 1.1: I get it the basket on the first throw!
• outcome 1.2: I miss the basket, but get the disc in the basket on the second throw.
• outcome 1.3: I miss twice, and get it on the third throw.

The probabilities for each of those outcomes are:  $p$, $(1-p) q$, and $(1-p)(1-q)$ respectively.

Let $a$ be the average number of throws for choice 1. Then \begin{aligned}a &= p\cdot 1 +(1-p)q\cdot 2 + (1-p)(1-q)\cdot 3 \\&= p + 2 q – 2 p q + 3 – 3 p – 3 q + 3 p q\\&=3 -2 p – q + p q.\end{aligned}

I should choose choice 1 if $2>a$.  This occurs when

\begin{aligned} 2 &> 3 -2 p – q + p q\\-1 &> -2 p – q + p q \\-1 &> (q-2) p – q \\q -1 &> (q-2) p\\ \frac{q -1}{q-2} &< p \\\frac{1-q}{2-q} &< p. \\ \end{aligned}

Now you can plug in various values for $q$ to find the lowest value for $p$ needed to go for it.

Probability of Success          Required
After Missing             Probability of Success
on the first try
100%                           0%
99%                           1%
95%                           5%
90%                           9%
85%                          13%
80%                          17%
75%                          20%
50%                          33%
0%                           50%

So, if you are 100% certain that you will put it in the basket on the second try, then you should use choice 1 (going for it) if $p>0$ (i.e. always).

If you are 90% certain that you will put it in the basket on the second try, then you should use choice 1 (going for it) if $p>0.09=9\%$.



### a rule of thumb

A nice approximate rule of thumb is to go for it if the sum of $p$ and $q$ is more than 100%.

When I am 6 yards from the basket, I will get it in about 75% of the time (p=0.75), and if I miss, I will usually get it in 90% of the time.  The sum of 70% and 90% is 160%, so obviously, I should go for it.

When I am 9 yards from the basket, I will get it in about 20% of the time (p=0.20), and if I miss, I will usually get it in 85% of the time.  The sum of 20% and 85% is 105%, so it seems like I should go for it.

If the basket is on a hill or if it is windy, then the disc will sometimes roll a fair distance if I miss.  In that case, $q$ might be only 75%. The rule of thumb is that $p+q$ should be at least 100% to go for it, so according to the rule of thumb, I would need $p$ to be at least 25% to go for it.  On a windy day, that’s like 6 yards for me.

## Achieving Immortality with GPT-3

GPT-3 is a rather amazing neural net. If you give it any snippet of English text, then it will try it’s best to continue the text.  Stephan Marshe at The Newyorker gave GPT-3 the introductory text of Kafka’s “The Metamorphis” which begins with,

“One morning, when Gregor Samsa woke from troubled dreams, he found himself transformed in his bed into a horrible vermin.”,

and continues with Gregor’s struggles with his parents and boss over the next few hours. At the end of this introduction, Gregor is in his bed. GPT-3 continues the story with

“As soon as Gregor was alone, he began to feel ill. Turning around was an effort. Even breathing was an effort. A thin stream of blood trickled from his flank down his fuzzy belly. He wanted to crawl away from it, but there was no place to go. He lay still on the spot where he had come to rest just in order to get his breath back and to stop the bleeding. “I’m in a bad way,” said Gregor. It had never occurred to him before that he could really become ill. He had seen sick animals—a dove once in a while, which had fallen out of the nestling into the gutter and could not fly any more, or the weak infants of the woman next door who had to be picked up with the tongs and thrown into the dustbin, or the bugs his father used to bring to him when he was still a young boy and which he had liked so much.”
The point is that GPT-3 can generate very realistic completions of any input text. When Kevin Laker gave GPT-3 the following input,

Q: What is human life expectancy in the United States?
A: Human life expectancy in the United States is 78 years.

Q: Who was president of the United States in 1955?
A: Dwight D. Eisenhower was president of the United States in 1955.

Q: What party did he belong to?
A: He belonged to the Republican Party.

Q: Who was president of the United States before George W. Bush?
A: Bill Clinton was president of the United States before George W. Bush.

Q: Who won the World Series in 1995?
A: The Atlanta Braves won the World Series in 1995.

Q: What are two reasons that a dog might be in a bad mood?

GPT-3 gave the response “Two reasons that a dog might be in a bad mood are if it is hungry or if it is hot.”
Basically, it is possible to carry on a conversation with GPT-3 by initializing it with the beginning of a conversation.

So, here is how you achieve a kind of immortality. You record all of your conversations for a year or two and use a speech-to-text converter to convert all of that speech into text while labeling the speakers. For example, if you have a conversation with your friend Mary, then you would enter in text similar to

START CONVERSATION

Mary: Hey Joe, how are you?
Me: I’m doing all right.
Mary: What have you been up to?
Me: I just got back from Denver yesterday. Tim and I were working on the robotic arm all week, and it seems to be getting better at folding the clothing.
Mary: ….

END CONVERSATION
“.

You could do this for every conversation that you had.

So now if Elon Musk (one of the founders of Open AI which developed GPT-3) wants to simulate a conversation with you, he could take that long transcript of your conversations and append

START CONVERSATION
Elon: Hey Joe, I haven’t seen you for a while, where have you been?
Me:
“.

Then GPT-3 would type out your most likely response to that question. Once GPT-3 reaches the end of your simulated response, then Elon could append his next contribution to the conversation and once again GPT-3 would generate your mostl likely response to Elon. In this way, GPT-3 could create a simulation of conversation with you.

This simulator could be improved by tweaking the parameters of the neural net to better fit your conversational style. You could also feed it many conversations between all kinds of people all prefaced with the bios of the each speaker. By giving GPT-3 more conversational text data and associated bio’s, the neural net could become significantly better at simulating people.

So, if you have a large collection of your own written text and you record yourself for a year, GPT-3 can create a simulation of you. This is a kind of immortality because the GPT-3 program and your conversational text can produce simulated conversations with you even after your death. These simulated conversations could become even more accurate if GPT-3 is improved further.

(One of my friends informed me that this ideas has been discussed on Reddit and the Black Mirror.)