Solutions to Practice Problems for  the Lander Math Contest
Module 14, Solutions (43-45)

43)
The expected value of the set of numbers is
E(X)=(3*1+2*5+4*6+1*7)/10=4.4 .

44) Let us calculate frequencies of getting 0,1,2 and 3 heads:
P(0)=1*1/2^3 = 0.125
P(1) =3*1/2^3 = 0.375
P(2) =3*1/2^3 = 0. 375
P(3) =1*1/2^3 = 0.125
Then, the mean number of heads will be:
E(X)=P(0)*0+ P(1)*1+ P(2)*2+ P(3)*3=
=0.125*0+0.375*1+0.375*2+0.125*3=1.5
QUESTION: Do you understand why 1.5 is the logical answer?

45) We count frequencies of receiving 0,1,2,3 and 4 heads:
P(0)=1*1/2^4 = 0.0625
P(1) =4*1/2^4 = 0.25
P(2) =6*1/2^4 = 0. 375
P(3) =4*1/2^4 = 0.25
P(4) =1*1/2^4 = 0.0625
Now, the mean number of heads is equal to:
E(X)=P(0)*0+ P(1)*1+ P(2)*2+ P(3)*3+ P(4)*4=
=0.0625*0+0.25*1+0.375*2+0.25*3+0.0625*4=2
QUESTION: Do you understand why 2 is the logical answer?