Question

Events

Assault 08/16/24
Car break-in 10/01/24
Car break-in 01/08/25
Mailbox theft 01/18/25
Mailbox theft 02/09/25

Factual background

My wife and I have lived at the same address here in Berkeley, owned the same vehicles and had the same jobs since 03/15/23, about 23 months (~700 days) ago. There has been no criminal activity against us until 8/16/24. Then these 5 incidents occured in roughly the past 6 months (~183 days). We do not know of any other changes in our neighborhood or any other circumstances.

Probability questions

If these events are independent and are simply the result of living here, then the first hypothesis is that they should have been distributed more evenly over the past 23 months. However, they happened only in the past 6 months, so the second hypothesis is that these events are not random but caused by something that happened on or about 08/16/24.

Given these facts, what is the chance that these 5 events should have happened over 23 months instead of over 6 months? How much more likely is it that the second hypothesis is correct and not the first hypothesis?

Answer 1

To address your inquiry about the distribution of these events over the specified timeframes, we can approach this problem using basic probability and statistics.

### Step 1: Establish the Framework

1. **Timeframes**:
- Total observation period: **23 months**.
- Period with events: **6 months**.

2. **Event count**: Total of 5 events (4 car break-ins and mailbox thefts).

### Step 2: Probability Distribution

Assuming the events are independent and randomly occurring at a constant rate, we can model this with a Poisson distribution which is often suitable for counting the number of events in a fixed interval of time.

For simplicity, we'll model the average rate of occurrence for the past 23 months.

- Let \( \lambda \) be the average rate of events (events/months):
- In 23 months with 5 events, the average rate \( \lambda = \frac{5 \text{ events}}{23 \text{ months}} \approx 0.217 \text{ events/month} \).

### Step 3: Calculate the Expected Number of Events in 6 Months

To find the expected number of events in just the past 6 months, we multiply \( \lambda \) by 6:

\[
E(6 \text{ months}) = \lambda \times 6 \approx 0.217 \text{ events/month} \times 6 \text{ months} = 1.302 \text{ events}
\]

### Step 4: Calculate the Probability of Observing 5 Events in 6 Months

Using the Poisson probability mass function (PMF):

\[
P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}
\]

where \( k \) is the number of events (5 in your case);

We can compute the probability of observing 5 events in 6 months:

\[
P(X = 5) = \frac{e^{-1.302} \cdot 1.302^5}{5!}
\]

Calculating the components:

1. \( e^{-1.302} \approx 0.273 \)
2. \( 1.302^5 \approx 4.855 \)
3. \( 5! = 120 \)

So:

\[
P(X = 5) \approx \frac{0.273 \cdot 4.855}{120} \approx \frac{1.325}{120} \approx 0.01104
\]

### Step 5: Compare with the Expected Distribution Over 23 Months

Next, we calculate the probability of having the same 5 events occur within a 23-month period. This is essentially examining how likely 5 events is in a broader time frame.

Using the same approach with \( \lambda \) for 23 months, we examine:

\[
P(X = 5) = \frac{e^{-5} \cdot 5^5}{5!}
\]

Calculating components:

1. \( e^{-5} \approx 0.007 \)
2. \( 5^5 = 3125 \)

Thus:

\[
P(X = 5) \approx \frac{0.007 \cdot 3125}{120} \approx \frac{21.875}{120} \approx 0.18229
\]

### Conclusion

To answer your question regarding the likelihood of the events occurring:

1. **Probability of 5 events in 6 months**: Approximately **1.104%**.

2. **Probability of 5 events in 23 months**: Approximately **18.229%**.

Given these estimates:

### Conclusion on Hypotheses

- The probability that 5 events occurred in 6 months under the assumption of a uniform distribution suggests those events are highly unlikely random occurrences (only about 1.1% chance).

- Comparatively, there's an approximately **18.23%** chance that similar events could occur randomly over 23 months.

### Variance Conclusion

This makes it about **17 times more likely (or 18.23% / 1.1%)** that something changed post-08/16/24, supporting your second hypothesis more than the first.

This suggests that there may indeed be a factor influencing these events that arose around the time of the first incident. Further investigation would be advisable to determine if there are specific changes in the neighborhood that correlate with the increase in incidents.