Let S be a set of 31 equally spaced points on a circle centered at O, and consider a uniformly random pair of distinct points A and B (A,B∈S). The probability that the perpendicular bisectors of OA and OB intersect strictly inside the circle can be expressed as mn, where m,n are relatively prime positive integers. Find m+n
I think you mean probability is m/n, where m≤n.
yes
is it 5
To find the probability that the perpendicular bisectors of two random points, A and B, intersect strictly inside the circle, we need to determine the favorable outcomes divided by the total number of possible outcomes.
First, let's determine the total number of possible outcomes. To do this, we calculate the total number of ways to choose two distinct points from the set of 31 points, which is denoted as 31C2 or "31 choose 2." This can be calculated using the binomial coefficient formula:
31C2 = 31! / (2!(31-2)!) = 31! / (2!29!) = (31 * 30) / 2 = 15 * 31 = 465
So, there are 465 total possible outcomes.
To determine the favorable outcomes, we need to find the number of pairs of points where the perpendicular bisectors of OA and OB intersect strictly inside the circle. We can analyze this geometrically.
Consider point A, which can be any of the 31 points on the circle. If we draw the perpendicular bisector of OA, it passes through the center O and divides the circle into two arcs. We need to find the range of points for B that results in the perpendicular bisectors intersecting strictly inside the circle. Let's analyze each case:
Case 1: If point B lies on the same half of the circle as point A (either both on the top half or both on the bottom half), the perpendicular bisectors of OA and OB intersect outside the circle. In this case, the outcome is not favorable.
Case 2: If point B lies on the opposite half of the circle compared to point A, the perpendicular bisectors of OA and OB intersect inside the circle. In this case, the outcome is favorable.
There are 15 points on each half of the circle (excluding point A itself), so the number of favorable outcomes in Case 2 is 15.
Since there are 31 possible choices for point A, the total number of favorable outcomes is 31 * 15 = 465.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total possible outcomes:
Probability = (Number of favorable outcomes) / (Total possible outcomes) = 465 / 465 = 1.
Therefore, the probability that the perpendicular bisectors of OA and OB intersect strictly inside the circle is 1.
To express this as a fraction, m/n, the sum of m+n is 1+1=2.