How confident can Little Johnny be in Confidence Intervals??

[JinX11]

Cross-posted in Probability; no love, yet.

I should understand this; I don't. I'm interested for non-poker related reasons, but I can express my problem in poker terms. Suppose the following scenario; someone correct me when I go off track.

Little Johnny starts playing poker. Over his first 7,000 hands, he enjoys a terrific run of cards, winning at a 7BB/100 clip with a SD of 17 BB/100. Though Little Johnny rarely paid attention in class and routinely teased the teacher, he knows a little about statistics and thinks he knows how to calculate the range of his true win rate to a certain degree of confidence with Excel.

Using an alpha of 0.05 (95% confidence), he determines that with his sample size (7000 hands, or 7000/100 = 70 samples), present win rate (7 BB/100), and standard deviation 17 BB/100, little Johnny asserts that he is 95% confident that his maximum win rate is 10.98 BB/100 and his minimum win rate is 3.02 BB/100. Consequently, he quits his job and decides to play poker for a living.

We all know that his sample size is too small, but this scenario can easily occur. If anything, we are virtually 100% confident that his true win rate does NOT lie between 3.02 and 10.98 BB/100. Where did Little Johnny go wrong?


Thanks, in advance, for any responses. Any help is appreciated.