Question

一道大学的数学题求解。！！！

路边有个卖东西的，路人买东西的概率是1/3，请问当卖出第100个东西的时候，路过的人数在280-300人之间的概率？
最好有过程，应该是相关二项分布的题目，求解，谢谢/

Answer 1

(1)
using the coordinate system as described,
the path of the ball is determined by
y=Vsinα·t-1/2·g·t^2
x=Vcosα·t

(2)
rotate the original coordinate system by an angle β to establish a new one, which preserves the orgin but uses the inclined plane as the X axis
based on results from question 1, we have the new path of the ball as:
y'=Vsin(α-β)·t-1/2·gcosβ·t^2
x'=Vcos(α-β)·t-1/2·gsinβ·t^2
when the ball lands on the plane, y'=0
hence
t=2Vsin(α-β)/(gcosβ)
(3)
further to the results of question 2, when the ball lands on the inclined plane,
x'=Vcos(α-β)·2Vsin(α-β)/(gcosβ)-1/2·gsinβ·[2Vsin(α-β)/(gcosβ)]^2=2·V^2·[sin(α-β)cos(α-β)-sin(α-β)·sin(α-β)tgβ]/(gcosβ)
x' reaches its maximum when its derivative equals to zero
ie. derivative of sin(α-β)cos(α-β)-sin(α-β)·sin(α-β)tgβ equals to zero.
Hence cos[2(α-β)]-sin[2(α-β)]tgβ=0
ctg[2(α-β)]=tgβ
α-β=(90-β)/2
That is to say, maximum range is achieved when α bisects the angle between the plane and the vertical

By plugging α-β=(90-β)/2 into the x' formulation,
you get the maximum.