The March 22, 2019 Riddler asks us to simulate baseball using probabilities from a 19th century dice game. There were some choices to make that were left unspecified in the rules; the following are my current choices (in an early version I made different choices that resulted in slightly more runs):

- On a
`b`

-base hit, runners advance`b`

bases, except that a runner on second scores on a 1-base hit. - On an "out at first", all runners advance one base.
- A double play only applies if there is a runner on first; in that case other runners advance.
- On a fly out, a runner on third scores; other runners do not advance.
- On an error all runners advance one base.
- On a base on balls, only forced runners advance.

I also made some choices about the implementation:

- I wanted to have one event per batter, so I don't allow "strike" as an event. Rather I compute the probability of a strikeout event (i.e. getting three "strike" dice rolls in a row before getting another event) as
`(7/36)**3`

, and check for that. - Note that a die roll such as (1, 1) is a 1/36 event, whereas (1, 2) is a 2/36 event, because it also represents (2, 1).
- I'll represent events with the following one letter codes:
`K`

,`O`

,`o`

,`f`

,`D`

: strikeout, foul out, out at first, fly out, double play`1`

,`2`

,`3`

,`4`

: single, double, triple, home run`E`

,`B`

: error, base on balls

- I'll keep track of runners with a list of occupied bases;
`runners = [1, 2]`

means runners on first and second. - A runner who advances to base 4 or higher has scored a run (unless there are already 3 outs).
- The function
`inning`

simulates a half inning and returns the number of runs scored. - I want to be able to test
`inning`

by feeding it specific events, and I also want to generate many innings worth of random events. So I'll make the interface be that I pass in an iterator of events. - I'll random simulate 1 million innings and store the resulting scores in
`innings`

. - To simulate a game I just sample 9 elements of
`innings`

and sum them.

In [1]:

```
%matplotlib inline
import matplotlib.pyplot as plt
import random
```

In [2]:

```
def our_national_ball_game():
"An iterator of events sampled from the odds specified in `Our National Ball Game`."
events = '2111111EEBBOOooooooofffffD334'
while True:
yield 'K' if random.random() < (7 / 36) ** 3 else random.choice(events)
def inning(events=our_national_ball_game(), verbose=False) -> int:
"Simulate a half inning based on events, and return number of runs scored."
outs = runs = 0 # Inning starts with no outs and no runs,
runners = [] # ... and with nobody on base
while True:
x = next(events)
if verbose: print(f'outs: {outs}, runs: {runs}, runners: {runners}, event: {x}')
if x in 'KODof': # strikeout, foul out, double play, out at first, fly out,
outs += 1 # Batter is out
if x == 'D' and 1 in runners: # double play
outs += 1
runners = [r + 1 for r in runners if r != 1]
elif x == 'o': # out at first (other runners advance)
runners = [r + 1 for r in runners]
elif x == 'f' and 3 in runners and outs < 3: # fly out; runner on 3rd scores
runners.remove(3)
runs += 1
else:
runners.append(0) # Batter becomes a runner
if x in '1234': # single, double, triple, homer
runners = [r + int(x) + (r == 2) for r in runners]
elif x == 'E': # error
runners = [r + 1 for r in runners]
elif x == 'B': # base on balls
runners = [r + all(b in runners for b in range(r)) for r in runners]
# See if inning is over, and if not, whether anyone scored
if outs >= 3:
return runs
runs += sum(r >= 4 for r in runners)
runners = [r for r in runners if r < 4]
```

Let's peek at some random innings:

In [3]:

```
inning(verbose=True)
```

Out[3]:

In [4]:

```
inning(verbose=True)
```

Out[4]:

And we can feed in any events we want to test the code:

In [5]:

```
inning(iter('2EBD12f'), verbose=True)
```

Out[5]:

That looks good.

Now, simulate a million innings, and then sample from them to simulate a million nine-inning games:

In [6]:

```
N = 1000000
innings = [inning() for _ in range(N)]
games = [sum(random.sample(innings, 9)) for _ in range(N)]
```

Finally, display the mean number of runs scored per team per nine-inning game, along with a histogram:

In [7]:

```
plt.hist(games, ec='black', bins=max(games)-min(games)+1)
sum(games) / N
```

Out[7]: