Poisson Calculator
Poisson takes one expected average and returns how often every count around it actually lands — which is what a total is really priced on.
An average of 2.70 puts 3 or more on the board 50.6% of the time, so the fair price is -103 on Over 2.5 and +103 on the under. That is the price with no fee in it — a book has to pay better than -103 before the over is worth taking.
If the average clears the line, the over is the obvious side.
The rate
What the rate spreads into
The exact count is the narrowest of these and almost never the bet. The over settles on the at-least row and the under on the fewer-than row — the two brightened here, and the two the fair price below comes off.
What that is worth as a price
These are fair prices, carrying no fee at all. A posted number has to pay better than the side you want before there is anything in the bet for you.
Every count, and how often
Each bar is that count against the most likely count, not against the whole board. Your target is the brightened row.
The rule underneath
One rate in, the whole spread of counts out. The page walks it step by step — start at the chance of none, then multiply by λ over each successive count — so a distant target never overflows.
Poisson assumes events arrive independently at a steady rate, and sport does not oblige. A side that goes ahead stops pushing, a red card rewrites the second half, and a blowout piles on counts after the result stopped being in doubt. It also ties the spread of outcomes to the average itself, so for streakier stats the real tails are fatter than this page draws them. Everything here is downstream of the average you typed: move λ by a tenth and the fair price moves further than most edges are worth.