My longest winning streak was 17 times in a row. However, I should mention my sessions are very long as I consider one day to be session which might average 8 hours of blackjack play. To show how meaningless sessions are the year I had this winning streak I actually had a loss for the year of $371. This happened fairly early in my blackjack playing history and it has never happened again.
Now in one of my better playing years I had a streak of 22 wins out of 24 sessions in the early part of the year, and then later on in the same year another streak of 19 wins out of 22 sessions. That was a good year.
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