Birds on a Wire

Consider a wire of length 1 unit between two posts. Every morning $n$ birds land on it randomly with every point on the wire equally likely to host a bird. The interval from each bird to its closest neighbour is then painted.
Define $F(n)$ to be the expected length of the wire that is painted. You are given $F(3) = 0.5$.
Find the average of $F(n)$ where $n$ ranges through all odd prime less than a million. Give your answer rounded to 10 places after the decimal point.

To solve this problem using Python, we can start by implementing a function to calculate F(n) for a given value of n. Then, we can iterate over all odd prime numbers less than a million and calculate their average.

First, let’s define the function to calculate F(n):

“`python
def calc_expected_length(n):
if n == 1:
return 0

# Calculate the number of painted intervals for each bird
num_painted_intervals = n – 1

# Calculate the expected distance between each pair of adjacent birds
expected_distance = 1 / (n + 1)

# Calculate the expected length of the wire that is painted
expected_length = expected_distance * num_painted_intervals

return expected_length
“`

Next, we can iterate over all odd prime numbers less than a million and calculate the average of F(n):

“`python
import math

def average_F():
primes = [2] # Initializing with the first odd prime number

# Generate all odd prime numbers less than a million
for num in range(3, 1000000, 2):
is_prime = True

for divisor in range(3, int(math.sqrt(num)) + 1, 2):
if num % divisor == 0:
is_prime = False
break

if is_prime:
primes.append(num)

# Calculate the average of F(n) for all odd primes
total = 0
for n in primes:
total += calc_expected_length(n)

average = total / len(primes)

return round(average, 10)
“`
Finally, we can call the `average_F()` function to get the rounded average of F(n) for all odd primes less than a million.

“`python
result = average_F()
print(result)
“`

This will give the desired output rounded to 10 decimal places.

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