[Math] Coin in circle probability.

probability

In a common carnival game a player tosses a penny from a distance of about
5 feet onto the surface of a table ruled in I-inch squares. If the penny (i inch
in diameter) falls entirely inside a square, the player receives 5 cents but does
not get his penny back; otherwise he loses his penny. If the penny lands on the
table, what is his chance to win?

Now, here's the solution provided.

When we toss the coin onto the
table, some positions for the center of
the coin are more likely than others,
but over a very small square we can
regard the probability distribution as
uniform. This means that the proba-
bility that the center falls into any
region of a square is proportional to the
area of the region, indeed, is the area of the region divided by the area of the square. Since the coin is 3/8 inch in
radius, its center must not land within 3/8 inch of any edge if the player is to
win. This restriction generates a square of side 1/4 inch within which the
center of the coin must lie for the coin to be in the square. Since the proba-
bilities are proportional to areas, the probability of winning is (1/4)^2 = 1/16.
Of course, since there is a chance that the coin falls off the table altogether,
the total probability of winning is smaller still. Also the squares can
be made smaller by merely thickening the lines. If the lines are 1/16 inch
wide, the winning central area reduces the probability to (3/16)^2 = 9/256 or
less than 1/28.

Now I understand the solution till it says that the probability of winning is 1/16. But what is happening after that?

Thanks

Best Answer

  • If the penny doesn't land on the table, it's clearly not within a square, so this is a loss. The $1/16$ probability is conditional on the penny landing on the table. If you include a non-zero probability of the penny missing the table altogether, the probability of winning goes down.
  • The solution assumes ideal (infinitely thin) lines dividing the squares. The loss criterion is that the penny touches a line. If the lines are thicker, this reduces the size of the square that the penny must land on, and reduces the winning probability.
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