Question

In a group of 100 sports fans, 90 are football fans and 10 are baseball fans. You have with you a box containing 90 football jerseys and 10 baseball jerseys. You distribute one jersey to each fan from this box in a random order. What is the expected number of baseball fans who get baseball jerseys? Show your work. (Hint: you can do an exact counting argument but this is cumbersome, try using linearity of expectation instead.)

Answer 1

here let Xi is indicator variable for which Xi =1 if baseball fan gets the baseball jersey

and Xi=0 ;if baseball fan did not receive baseball jersey

for Xi=1 ; probability =10/100 =0.1 (as there are 10 baseball jersey among 100)

total number of baseball fans who get baseball jerseys Y=X1+X2+X3+X4+..+X10

hence expected number of baseball fans who get baseball jerseys

E(Y)=E(X1)+E(x2)+E(X3)+..+E(X10)=0.1+0.1+0.1+0.1+..+0.1 =10*0.1 =1