16. Bridges. (From 9th grade, 3 points) Along the southern shore of an endless sea, there stretches an archipelago of an infinite number of islands. The islands are connected by an infinite chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independently of the others is destroyed with a probability of $p=0.5$. What is the probability that after a strong earthquake, one can cross from the first island to the shore using the surviving bridges?
Solution. Let's number the bridges as shown in Figure 11. If bridge $b_{1}$ remains intact (the probability of this is $q=1-p$), then one can cross to the shore using it.

![](https://cdn.mathpix.com/cropped/2024_05_06_11f8b78b86004d31b3e4g-12.jpg?height=437&width=1030&top_left_y=1598&top_left_x=593)

Fig. 11.

If bridge $b_{1}$ is destroyed, but bridges $a_{1}$ and $b_{2}$ remain intact, then one can cross to the shore using them. The probability of this event is $p q^{2}$.

And so on: if all bridges from $b_{1}$ to $b_{n}$ are destroyed, but bridge $b_{n+1}$ remains intact, then one can cross to the shore only if bridges $a_{1}, a_{2}, \ldots, a_{n}$ remain intact. The probability of such an event is $p^{n} q^{n+1}$.

Since there are infinitely many islands, there are also infinitely many such events, and all of them are mutually exclusive. The total probability of the event "one can cross to the shore from the first island using the bridges" is the sum of the probabilities of these events:

$$
q + p q^{2} + p^{2} q^{3} + \ldots + p^{n} q^{n+1} + \ldots = \frac{q}{1 - p q}
$$

Substituting $p = q = 0.5$, we find the desired probability: $\frac{0.5}{1 - 0.25} = \frac{2}{3}$.

Answer: $2 / 3$.