How likely are you to find a message in a bottle?
Recently, a happy century-old message in a bottle was found on the southwest coast of Australia. In it, a World War I soldier proclaimed himself “as happy as Larry.”
If you’re a betting man, you probably don’t expect a great chance of this happening. A bottle thrown into the ocean can end up anywhere.
So what are the chances that a message in a bottle will be found and it will be greater than 100? And what are your chances of finding this bottle?
Despite these many possibilities over the life of a bottle, the probability we are looking for is a simple calculation. Simply count the number of message bottles that have been found that are over 100 years old, and divide them by the number of messages that have been sent this way (assuming we know how many are sent):
Our diagram below shows a hypothetical situation in which 20 bottles are sent in total, of which six are found (indicated in gold) and one of which is over 100 years old (indicated by the “100” stamp). Thus, one bottle in 20 is found and is more than 100 years old. (Note: This is just a hypothetical calculation, not actual data.)
Instead of calculating the probability directly, another way to do this is to split the problem into two parts: (A) a bottle with a message is found, and (B) the bottle found is greater than 100. These two probabilities can be calculated separately and multiplied together to get what we want:
This is called the “multiplication rule” of probability, and we confirm from our hypothetical numbers that (6/20) × (1/6) = 1/20, as before.
Both approaches to calculating this probability are simple. However, direct calculation requires knowing the total number of bottles shipped, which is very difficult to know in the real world.
The multiplication rule has the advantage of dividing the calculation into two parts. We can address each separately and then put the two results together to get the desired probability. This is useful in real world situation where we can extract information from different sources.
First, we will deal with the probability that a bottle with a message will be found, regardless of its age.
Experts from the Federal Maritime and Hydrographic Agency of Germany suggests a 1 in 10 chance that a message in a bottle will be found. This largely corresponds to various historical “drift bottle” experiments, in which oceanographers released large numbers of bottles to understand ocean currents.
For example, studies conducted in the 1960s and 1970s in the North Atlantic Ocean led to recovery rates of 14% of the Gulf of Mexico, 8% of the Caribbean Sea And 7% of the northern coast of Brazil. A more recent and more northern study (between Canada and Greenland) dating from the 2000s led to a 5% recovery rate.
We would expect results to naturally vary across different experiments conducted in different parts of the world. But to keep things simple, we’ll stick with 1/10 as the probability that a bottle with a message will be found.
Let’s now move on to the second element of the calculation: among the bottles found, what proportion is over 100 years old?
The table below summarizes data from news articles collected on Wikipedia on very old bottles with found messages. However, only data on bottles older than 25 years was collected, probably because older bottles are more worthy of interest.
We therefore had to estimate ourselves the number of bottles from 0 to 25 years old with messages. Here’s how we did it.
The table shows that fewer bottles with messages are found as they age. The messages in the bottles degrade over time, meaning the bottles have an increased risk of breaking and leaking, or simply becoming covered in layers of sediment. Plotting this data in the graph below helped us see the trend in the age of the bottles found more clearly.
We drew a line to match this trend observed at the time of the found bottles. This red line in the graph corresponds to the equation:
This equation provides an estimate of the number of bottles found for a specific age range (where 25 = 0 to 25 years, 50 = 25 to 50 years, etc.). We are interested in bottles aged 0 to 25 years, so the equation suggests that 46 bottles were found in this range.
Adding this and all the numbers in the table, we get a total of 106 bottles found, 12 of which are over 100 years old, and 12/106 is about one in ten.
Summarizing the above, we have that: (A) one in ten bottles with messages are found, of which (B) one in ten is over 100 years old. Putting these results together using the multiplication rule, we estimate the chance that a message in a bottle is found and is over 100 years old at (1/10) × (1/10) = 1/100.
So if there are 100,000 bottles with messages floating in the oceans waiting to be found, we would expect 1,000 of them to be found and to be 100 years old or older. Assuming that anyone in the world has the same chance of finding one, with 8 billion people currently, that’s about a 1 in 8 million chance that you’ll find one – which is pretty unlikely.
However, some people are more persistent hunting for the message in a bottle than the others. By following the paths of ocean currents (called gyres) might provide clues on where to look.
Specifically, peninsulas or islands that intersect these gyres could be good locations. For this reason, it was suggested The Caribbean islands are ideally placed for finding bottles while they are in the path of the North Atlantic Gyre. Which sounds like a great reason to travel to the Caribbean!
But let’s also think about the poor souls stranded on their desert island, who will surely not appreciate the slim chances that their SOS will be found.
This edited article is republished from The conversation under Creative Commons license. Read the original article.
