F.3 Maths - Probability 20 分

20a)
Let x be the side length of the orange square
Then the radius is √((x/2)^2+(x/2)^2)=x/√2

P(hit the orange region)
=The orange area/The total area
=x^2/pi*(x/√2)^2
=1/(pi*(1/2))
=2/pi

b)
Expected value
=1*P(hit the orange region)+3*P(hit the blue region)
=1*(2/pi)+3*(1-(2/pi))
=(3pi-4)/pi marks

21a)
Let AD=x
Area MNC=x^2-(1/2)(x/2)^2-(1/2)(x)(x/2)*2=3x^2/8

P(hit MNC)
=area MNC/total area
=(3x^2/8)/x^2
=3/8

b)
Expected value
=5*P(hit MNC)+(-3)*P(hit other region)
=5*(3/8)+(-3)*(1-3/8)
=0