Question

When solving for the probability why do I have to multiply the individual probabilities and then add them?

Let X = B(3,2/3) and Y = B(4,½). Compute P(X = Y).

Answer 1

here since events X and Y are independent

therefore from independence rule P(X=x, Y=y) =P(X=x)*P(Y=y)

also since 4 ways when both X and Y are 0,1,2,3 , X and Y are equal, therefore we add these 4 probability.

P(X=Y) =P(X=0,Y=0)+P(X=1,Y=1)+P(X=2,Y=2)+P(X=3,Y=4)

=P(X=0)*P(Y=0)+P(X=1)*P(Y=1)+P(X=2)*P(Y=2)+P(X=3)*P(Y=3)

=(3C0)*(2/3)^0*(1/3)^3*(4C0)*(1/2)^0*(1/2)^4+(3C1)*(2/3)^1*(1/3)^2*(4C1)*(1/2)^1*(1/2)^3+(3C2)*(2/3)^2*(1/3)^1*(4C2)*(1/2)^2*(1/2)^2+(3C3)*(2/3)^3*(1/3)^0*(4C3)*(1/2)^3*(1/2)^1

=0.298611