1. Introduction

Knowing that 200 undersea cables break every year globally, we estimate the probability of:

• A single cable break in the Baltic Sea on a given day.
• Three cable breaks in the Baltic Sea on the same day.

These estimates are based on global statistics and reasonable approximations.

2. Average Daily Cable Breaks Globally

• Total global undersea cable breaks per year: 200.
• Number of days in a year: $365$.
• Average breaks per day globally:

$\frac{200}{365} \approx 0.548 \, \text{breaks/day}$

3. Estimating Cable Breaks in the Baltic Sea

• Assuming the Baltic Sea contains about $1\%$ of the world’s undersea cables:

$\text{Expected breaks/year in the Baltic Sea: } \allowbreak 0.01 \times 200 = 2 \, \text{breaks/year}.$

• Average breaks per day in the Baltic Sea:

$\frac{2}{365} \approx 0.00548 \, \text{breaks/day}.$

4. Using the Poisson Distribution

Since cable breaks are rare and occur independently, we use the Poisson distribution with rate $\lambda = 0.00548$ (average number of daily breaks in the Baltic Sea).

The probability mass function is:

$P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$

where $k$ is the number of events.

4.1 Probability of a Single Cable Break ($k = 1$)

For $k = 1$ and $\lambda = 0.00548$:

$P(1; 0.00548) = \frac{e^{-0.00548} \cdot 0.00548^1}{1!}$

Steps:

• $e^{-0.00548} \approx 1 - 0.00548 = 0.99452$.
• $P(1; 0.00548) = 0.99452 \cdot 0.00548 \approx 0.00545$.

Thus, the probability of a single break is approximately:

$P(1; 0.00548) \approx 0.00545 \, (\text{or about } 0.545\%).$

4.2 Probability of Three Cable Breaks ($k = 3$)

For $k = 3$ and $\lambda = 0.00548$:

$P(3; 0.00548) = \frac{e^{-0.00548} \cdot 0.00548^3}{3!}$

Steps:

• $e^{-0.00548} \approx 0.99452$.
• $0.00548^3 \approx 1.64566 \times 10^{-7}$.
• $3! = 6$.
• $P(3; 0.00548) = \frac{0.99452 \cdot (1.64566 \times 10^{-7})}{6} \allowbreak \approx 2.73 \times 10^{-8}$.

Thus, the probability of three breaks is approximately:

$P(3; 0.00548) \approx 2.73 \times 10^{-8}.$

Conclusion

• The probability of a single cable break in the Baltic Sea on a given day is approximately:

$P(1; 0.00548) = ~0.545\%.$

• The probability of three cable breaks in the Baltic Sea on the same day is about:

$P(3; ~0.00548) ~\approx~2.73~\times~10^{-8}~\allowbreak (\text{or approximately }~1~\text{in }~36~\text{million}).$