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Table balance

Table balance

A carpenter with eyes closed makes 4 random holes on a circular board. If a leg is inserted into each hole,

What is the probability that the table can stand in balance on its legs?

Solution

Let A, B, C and D be the four holes made on the circular board and O its center. In order for the table to NOT stand in balance, the four points must be on the same side of a diameter. This condition can be characterized as follows (the angles are always measured counterclockwise so that they will always adopt any magnitude between 0 and 360 degrees): For one of the points, for example the A, the AOB, AOC and AOD angles They are less than 180 degrees. The probability that each of these angles will verify the condition is 1/2, and that all three will verify it is 1/8. As it is enough that this happens for one of the points we must still multiply that amount by four, that is to say that we have 1/2 or what is the same, We have a 50% chance that the table can stand in balance.