DATA: 1. Bullseye (Target) measures 7" in diameter. 2. Axe blade width can measure between 1" and 4". 3. 51% of the Axe blade must be within the 7" bullseye (target). Question: What axe size (1" to 4") would have the better likely hood of hitting the bullseye area if your throwing skills where not as good as the next player who is using a 4" axe blade?
Should i use a smaller 1" axe blade or continue to use the 4" axe blade?
I hope i made some sense of this question.
Thank, Bill
We can model this with three variables: the coordinates $x,y$ where the center of the axe blade lands, and the angle of the blade from vertical. Actually, using the symmetry of the target we can reduce the problem to just two variables: the distance $r$ from the center of the bullseye to the center of the blade, and the angle the blade makes with the radial line from the center of the target to the center of the blade.
With a bullseye of diameter $7$ and a blade of width $4$, if $r \geq 3.5$ then the center of the blade is on or outside the edge of the bullseye and you have certainly missed; there is no way for $51$% of the blade to be inside the bullseye.
If $r < 3.5$ and if at least one end of the blade is inside the bullseye, then all parts of the blade between the center and that end also are inside the bullseye, and so is some additional part of the blade, so you have a hit. (I'm assuming in this paragraph that "$51$%" really means "more than half".)
If the center of the blade lands barely within the bullseye and almost perpendicular to the radial line, however, then it is possible that both ends of the blade are outside the bullseye. That is, the blade cuts the bullseye along a line such that the line meets the circular boundary of the bullseye at two points, each of which is closer to the center of the blade than half the blade's width.
Consider the case where the blade cuts the bullseye along a line such that the two points of intersection of the line and the circle are each less than $2$ inches from the center of the blade, but at least one of them is more than $1/2$ inch from the center of the blade. Then the $1$-inch blade is a hit while the $4$-inch blade is a miss.
This seems like an event of non-zero but still rather small probability, not to mention the question of how they measure things when the blade is partly inside the bullseye but nearly tangent to the edge. I would also consider whether your ability to get the center of the axe in the bullseye is better or worse with the smaller axe than the larger one.