What is the probability that an eastbound ship does not pass any westbound ships during its journey through the canal?
To determine the probability that an eastbound ship does not pass any westbound ships during its journey through the canal, we need some additional information. Specifically, we need to know the average number of westbound ships that pass through the canal within a given time period.
Once we have this information, we can compute the probability using the Poisson distribution. The Poisson distribution is a probability distribution that expresses the likelihood of a given number of events occurring within a fixed interval of time or space, given the average rate of occurrence.
Here's the step-by-step process to calculate the probability:
1. Determine the average rate of westbound ships passing through the canal within the desired time period. Let's say this value is denoted by λ (lambda).
2. Using the value of λ, we can calculate the probability of exactly zero westbound ships passing through the canal during the journey. This is given by the Poisson probability mass function (PMF) formula:
P(X = 0) = (e^(-λ) * λ^0) / 0!
where e is the base of the natural logarithm (approximately 2.71828) and 0! is the factorial of 0 (which equals 1).
3. Plug in the value of λ into the formula and calculate the probability. The result will be the probability that an eastbound ship does not pass any westbound ships during its journey through the canal.
Remember, to accurately determine the probability, it is crucial to have an appropriate estimate for the average rate of westbound ships passing through the canal within the specified time period.
To find the probability that an eastbound ship does not pass any westbound ships during its journey through the canal, we need to consider a few factors:
1. Assume there are a total of "n" ships traveling through the canal.
2. Since the order of ships matters, the total number of possible arrangements of ships is n!.
Now, let's break down the problem step-by-step:
Step 1: Determine the number of ways to choose the position of the eastbound ship in the sequence of ships.
Since the eastbound ship cannot pass any westbound ships, it can be placed in any of the (n-1) positions in the sequence of ships. Therefore, there are (n-1) ways to choose the position of the eastbound ship.
Step 2: Determine the number of ways to arrange the remaining (n-1) ships.
After placing the eastbound ship, there are (n-1) ships remaining to be arranged. The number of ways to arrange these ships is (n-1)!.
Step 3: Calculate the probability.
The probability that the eastbound ship does not pass any westbound ships is equal to the number of favorable outcomes divided by the total number of possible outcomes. In this case, the favorable outcome is when the eastbound ship is placed in a position where it does not pass any westbound ships. The total number of possible outcomes is the total number of arrangements of ships, which is n!.
Therefore, the probability is ((n-1)! / n!).
Simplifying this expression, we get:
Probability = 1/n
So, the probability that an eastbound ship does not pass any westbound ships during its journey through the canal is 1/n.