7.2 There are three ants on different vertices of a triangle. What is the probability of collision (between any two or all of them) if they start walking on the sides of the triangle? Assume that each ant randomly picks a direction, with either direction being equally likely to be chosen, and that they walk at the same speed. Similarly, find the probability of collision with n ants on an n-vertex polygon.
這道題說有一個三角形，每個頂點上有一隻螞蟻，如果它們隨機選一個方向勻速爬，問任意兩隻會碰撞的概率是多少？然後又擴充到n邊形，放n只螞蟻，還是求會碰撞的概率。
我剛開始還在考慮三角形是否為等邊三角形，其實跟這些都沒啥關係。任意兩隻螞蟻碰撞的概率不好求，我們可以反其道行之，先求出不會碰撞的概率，然後取個反也能達到目的。那麼不會有碰撞的概率只有兩種情況，所有螞蟻都順時針爬，或者都逆時針爬。那麼對於三角形來說，3只螞蟻都順時針爬的概率是0.53,都逆時針爬的概率也是0.53，則相同方向爬的概率是0.53+0.53=0.25，則會碰撞的概率為1-0.25=0.75。那麼對於n邊形和n只螞蟻也是同理，n只螞蟻都順時針爬的概率是0.5n,都逆時針爬的概率也是0.5n，則相同方向爬的概率是0.5n+0.5n=0.5n-1，則會碰撞的概率為1-0.5n-1