How Many Ordered Pairs Meet the Criteria for Ball Arrangement Probability?

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The discussion centers on determining the number of ordered pairs (b,y) of positive integers within the range 4 ≤ b ≤ y ≤ 2007, where the probability of the first and last balls being the same color equals 1/2. Participants suggest that this is a conditional probability problem, possibly framed as a homework challenge. There is also a query about a potential feature for simplifying LaTeX input, indicating a desire for efficiency in formatting mathematical expressions. The conversation highlights the mathematical intricacies involved in calculating the specified probabilities. Overall, the focus remains on the probability criteria for the arrangement of colored balls.
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How many ordered pairs (b,y) of positive integers with 4 \le b \le y \le 2007 are there such that when b blue balls and y yellow balls are randomnly arranged in a row, the probability that the balls on each end have the same colour is 1/2?

PS: Is there a button such that I could just click on it after I highlighted the text that I want to be included in the LaTeX tags? It's going to save me a lot of typing if there's such button (without keep typing and so on)!
 
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This just looks like some conditional probability problem. Is it homework, a challenge for the rest of us, or something else?
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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