1. May 2006 Question 2  The head circumference of 3-year-old boys is known to be Normally distributed with mean 49.7cm and standard deviation 1.6cm.  Find the probability of having a head (A) over 51.5cm (B) between 48cm and 51.5cm  4 3-year-old boys are selected. Find the probability that exactly one of them has a head size between 48cm and 51.5cm

2. May 2006 Question 2 Solution X~N(49.7,1.6 2 ) (i)(A) P(X>51.5)=P(Z>51.5)=P(Z> ) P(Z>1.125)=1-P(Z<1.125) =1-Φ(1.125) =1-0.8696 =0.1304

3. (i)(B) P(48≤X≤51.5)=P(X≤51.5)-P(X≤48) =P(Z≤1.125)-P(Z≤ ) = Φ(1.125)- Φ(-1.063) =0.8696-(1- Φ(1.063)) =0.8696-(1-0.8561) =0.8696-0.1439 =0.7257

4. (ii) P(48≤X≤51.5)=0.7257 n=4 Therefore X~Bi(4,0.7257) P(X=1)= =0.0599

5. Key Points  Look carefully at the question.  See if you can use anything you’ve calculated before.  Just because the binomial distribution is not on the S2 syllabus directly, doesn’t mean they won’t ask you to use it.