Binomial as Normal

Question

Chelsea play 38 league games in a season. Their probability of losing each game is 0.25. If they lose fewer than 8 games, they will win the league. What is the probability they win the league?


Remember the steps;

Should you?

Could you?

The mean and standard deviation.

The boundaries?

The z-scores?

The table.

The interpretation.

Answer 1

 

 X' is Binomial N=38, p=0.25 ---> q=0.75

 

 mean = Np = 38*0.25 = 9.5 > 5

 

 So 5 < 9.5 = 38*0.25 < 38*0.75 = Nq

 

So the normal distribution can be used to approximate the binomial.

The mean is , as stated, Np=9.5

the standard deviation is sqrt(Npq) = sqrt( 38*0.25*.75) = 2.669269563007827802703....

 

Prob (X' < 8) = Prob(X < 8.5) <--- continuity correction

 

                   = Prob (z < (8.5 - 9.5)/2.669269....) =

                   = Prob ( z < -0.37463) = 0.3538 = 35.38%