The 538 Riddler for March 22, 2019 asks us to simulate baseball using probabilities from a 19th century dice game called *Our National Ball Game*:

```
1,1: double 2,2: strike 3,3: out at 1st 4,4: fly out
1,2: single 2,3: strike 3,4: out at 1st 4,5: fly out
1,3: single 2,4: strike 3,5: out at 1st 4,6: fly out
1,4: single 2,5: strike 3,6: out at 1st 5,5: double play
1,5: base on error 2,6: foul out 5,6: triple
1,6: base on balls 6,6: home run
```

The rules left some things unspecified; 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:

- Exactly one outcome happens to each batter. We call that an
*event*. - 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

- Note the "strike" dice roll is not an event; it is only part of an event. From the probability of a "strike" dice roll, I compute the probability of three strikes in a row, and call that a strikeout event. Sice there are 7 dice rolls giving "strike", the probability of a strike is 7/36, and the probability of a strikeout is (7/36)**3.
- 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 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 random innings. So I'll make the interface be that I pass in an*iterable*of events. The function`event_stream`

generates an endless stream of randomly sampled events. - Note that it is consider good Pythonic style to automatically convert Booleans to integers, so for a runner on second (
`r = 2`

) when the event is a single (`e = '1'`

), the expression`r + int(e) + (r == 2)`

evaluates to`2 + 1 + 1`

or`4`

, meaning the runner on second scores. - I'll play 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 event_stream(events='2111111EEBBOOooooooofffffD334', strike=7/36):
"An iterator of random events. Defaults from `Our National Ball Game`."
while True:
yield 'K' if (random.random() < strike ** 3) else random.choice(events)
def inning(events=event_stream(), 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
for e in events:
if verbose: print(f'{outs} outs, {runs} runs, event: {e}, runners: {runners}')
# What happens to the batter?
if e in 'KOofD': outs += 1 # Batter is out
elif e in '1234EB': runners.append(0) # Batter becomes a runner
# What happens to the runners?
if e == 'D' and 1 in runners: # double play: runner on 1st out, others advance
outs += 1
runners = [r + 1 for r in runners if r != 1]
elif e in 'oE': # out at first or error: runners advance
runners = [r + 1 for r in runners]
elif e == 'f' and 3 in runners and outs < 3: # fly out: runner on 3rd scores
runners.remove(3)
runs += 1
elif e in '1234': # single, double, triple, homer
runners = [r + int(e) + (r == 2) for r in runners]
elif e == 'B': # base on balls: forced runners advance
runners = [r + forced(runners, 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]
def forced(runners, r) -> bool: return all(b in runners for b in range(r))
```

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('2EBB1DB12f', verbose=True)
```

Out[5]:

That looks good.

Now, simulate a million innings, and then sample from them to simulate a million nine-inning games (for one team):

In [6]:

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

Let's see histograms:

In [7]:

```
def hist(nums, title):
"Plot a histogram."
plt.hist(nums, ec='black', bins=max(nums)-min(nums)+1, align='left')
plt.title(f'{title} Mean: {sum(nums)/len(nums):.3f}, Min: {min(nums)}, Max: {max(nums)}')
hist(innings, 'Runs per inning:')
```

In [8]:

```
hist(games, 'Runs per game:')
```