An equilateral triangle has one vertex at the centre of a disk.
One side of the triangle lies completely outside the disk and is colored green.
A red line is drawn through two independent, uniformly random points on the disk.
What is the probability that the red line crosses the green side?
1/3
We can add 5 more identical equilateral triangles with a vertex at the center of the circle, and their "green" sides would form a regular hexagon that surrounds the circle. Any line in the circle will almost surely pass through 2 of the triangles' green sides. Since it's equally likely that any 60 degree rotation about the center of that line would be randomly chosen, any particular green line will be intersected one third of the time.
Here’s a (cheeky) trick solution relying on the fact that
the radius of the disk isn’t given.
So we can
shrink the disk to a point (the apex of the triangle)
while still satisfying the condition that the green segment is entirely outside the disk. Now it’s clear that the red line
is chosen randomly from all the lines passing through the apex. When you rotate that line about the apex by 180° starting from any initial orientation, it goes back to the original orientation, and intersects the green segment exactly for a third of the orientations (because the green segment subtends an angle of 60° about the point of rotation, the apex).
Hence the answer is
1/3.
| 心脏彩超可以检查什么jingluanji.com | 平诊是什么意思hcv9jop0ns6r.cn | 微信密码是什么hcv8jop0ns8r.cn | jewelry什么意思hcv7jop7ns1r.cn | 空调干燥是什么意思hcv8jop8ns3r.cn |
| 智齿发炎是什么原因hcv7jop6ns2r.cn | 心跳过快有什么危害hcv9jop2ns5r.cn | 检查是否怀孕要挂什么科hcv8jop1ns0r.cn | 小便白细胞高是什么原因hcv9jop7ns5r.cn | 李倩梅结局是什么hcv9jop3ns9r.cn |
| 嘴唇发乌是什么原因hcv8jop8ns7r.cn | 大爷是什么意思hcv7jop5ns4r.cn | 胃黏膜病变是什么意思hcv8jop9ns6r.cn | 梗犬是什么意思sanhestory.com | 胸部爱出汗是什么原因hcv8jop6ns3r.cn |
| 胸导联低电压什么意思hcv9jop3ns6r.cn | 血糖突然升高是什么原因hcv8jop7ns1r.cn | 蛇进家里是什么预兆inbungee.com | 发烧不能吃什么水果hcv9jop8ns1r.cn | 弥散是什么意思hcv8jop0ns0r.cn |
