Expected Value of Even Numbers

Probability Level 3

Start by choosing two random natural numbers. Make a Fibonacci sequence starting with those two numbers. In other words, F 1 F_1 and F 2 F_2 are the two numbers you chose, and for n ≥ 3 n \geq 3 , F n = F n − 1 + F n − 2 F_n= F_{n-1} + F_{n-2} . What is the expected percentage of even numbers in the sequence? For clarification, I mean 100 × expected amount of even numbers amount of numbers in the sequence 100 \times \frac{\text {expected amount of even numbers}}{\text{amount of numbers in the sequence}} .

Write your answer as a decimal rounded to the nearest thousandth.


The answer is 50.000.

Jesse Li by Jesse Li , 16, USA

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1 solution

Geoff Pilling
Oct 22, 2018

You have a 1 4 \frac{1}{4} chance for each of the following initial combinations:

  • EE
  • EO
  • OE
  • OO

For the first one, all elements of the series are even.

However, for the other three you end up with the sequence:

...EOOEOOEOOEOOEOOE...

with exactly 1/3 of the numbers being even.

So, the expected percentage is:

E = 1 + 1 3 + 1 3 + 1 3 4 = 0.5 = 50 % E = \dfrac{1 + \frac{1}{3} + \frac{1}{3} + \frac{1}{3}}{4} = 0.5 = 50\%

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