The plane is ruled by parallel lines inches apart and a -inch long needle is dropped at random on the
plane. What is the probability that it hits parallel lines?
The first issue is to find some appropriate probability space . For this,
distance from the center of the needle to the nearest line
the angle that the needle makes with the horizontal ranging from to .
These fully determine the position of the needle. Let us next take the
The probability space is
The probability of an event is denoted by is equal to
Now we denote by the event that the needle hits a horizontal line. It is easily seen that this happens when . Consequently and then we get
In general case, when the length of needle is and the distance of parallel lines is provided that , the probability we want is . This is obvious just taking the -point from one edge instead of the center of the needle.
|Date of creation||2013-03-22 16:09:28|
|Last modified on||2013-03-22 16:09:28|
|Last modified by||georgiosl (7242)|