In a sequence of numbers 4 and 5 randomly arranged, let A be the number of numbers until the sequence 45 first appears, and B be the number of numbers until the sequence 55 appears. What is the magnitude relationship between the expected values of the numbers of A and B?

in Statistics Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

Best answer

Assuming numbers are restricted to positive integers, and that the integers are arranged in numerical order, the sequence up to 55 is:

4, 5, 44, 45, 54, 55.

A={4, 5, 44, 45}; n(A)=4

B={4, 5, 44, 45, 54, 55}; n(B)=6.

Relative magnitude n(B)/n(A)=6/4=3/2.

However, if the digits are arranged sequentially side by side in an infinite sequence, for example, 5454554544454545... then a different logic must be applied. In this example, 45 appears in the 2nd and 3rd position so A would be 3 and B would be 6. This would be analogous to placing coins in a line and counting how many coins are placed until a head is followed by a tail, and two tails first appear.

In any sequence of 4s and 5s in a long string there will always be more 45 permutations than 55. I estimate that there are about 40% more. But if you take any pair of digits it’s just as likely that the first appearance of 45 and 55 is the same probability, because there are 4 possible permutations:

44, 45, 54, 55 and the probability of 45 or of 55 is 1 in 4.

by Top Rated User (813k points)

Related questions

Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
85,985 questions
91,878 answers
2,227 comments
23,904 users